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Weak gravitational lensing cross-correlations Tröster, Tilman 2017

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Weak gravitational lensing cross-correlationsbyTilman Tro¨sterA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)August 2017c© Tilman Tro¨ster, 2017AbstractThe matter content of the Universe is dominated by dark matter. Beyond its abundance and its lackof non-gravitational interactions with standard model matter, little is known about the nature of darkmatter. This thesis attempts to illuminate different aspects of dark matter by using gravitational lensingin conjunction with other cosmological probes. Gravitational lensing describes the deflection of lightby gravitational potentials and is a direct and unbiased probe of the matter distribution in the Universe.We investigate the weakly interacting massive particle (WIMP) model of dark matter by per-forming a tomographic and spectral cross-correlation between weak gravitational lensing from theCanada-France-Hawaii Telescope Lensing Survey (CFHTLenS), Red Cluster Sequence Lensing Sur-vey (RCSLenS), and Kilo-Degree Survey (KiDS), and gamma rays from Fermi-LAT. The non-detectionof a correlation allows us to constrain the allowed masses, annihilation cross-sections, and decay ratesof WIMP dark matter.Even though most matter in the Universe is dark matter, about 16% is baryonic matter. To makeprecision measurements of the dark matter distribution with gravitational lensing, it is therefore nec-essary to understand the behaviour and distribution of baryonic matter. We measure and analysethe cross-correlation between weak gravitational lensing from RCSLenS and the thermal Sunyaev-Zeldovich (tSZ) effect from the Planck satellite to constrain the effect of baryon physics on the matterdistribution.Models of gravitational lensing often make use of a range of approximations. We study the effectof dropping these approximations on the cross-correlation between gravitational lensing and tSZ byperforming a detailed calculation up to fourth order in the gravitational potential. We find that thecommon approximations are sufficiently accurate even for future surveys.Finally, we explore the growth of structure and the effect of residual weak lensing systematics in atomographic cross-correlation between weak gravitational lensing from KiDS and lensing of the cosmicmicrowave background (CMB) from Planck.iiLay summaryThis thesis aims to illuminate various aspects of dark matter, a mysterious substance that makes up themajority of matter in the Universe. The existence and behaviour of dark matter can only be inferred fromits gravitational effect on its surroundings since it is impossible to observe directly. The distribution ofdark matter can be measured with gravitational lensing, which is caused by the deflection of light raysby matter in the Universe and can be observed by studying the distortion of the images of farawaygalaxies. Using gravitational lensing by itself and in conjunction with other observations, we shed lighton: whether dark matter could be a hitherto unobserved elementary particle; how the distribution ofdark matter compares to that of normal matter; and if the evolution of the dark matter distribution withtime is in agreement with that predicted by the standard model of cosmology.iiiPrefaceChapters 2 and 4 have been adapted from articles published in peer-reviewed journals, while Chapters 3and 5 have been adapted from submitted but not yet published articles. The articles on which Chapters2 and 5 are based were written as part of the KiDS collaboration, while the one on which Chapters 3is based on was written as part of the RCSLenS collaboration. Both collaborations have a publicationpolicy that breaks the author list into three groups: the first group consists of the primary authors whoconducted the main analysis and wrote the manuscript; the second group consists of members whoprovided infrastructure work on which the analysis was based and contributed significantly to the finalmanuscript; and the third group consists of members who provided infrastructure work or contributedsignificantly to the final manuscript.Chapter 2 is adapted from the article ‘Cross-correlation of weak lensing and gamma rays: implica-tions for the nature of dark matter’ by T. Tro¨ster, S. Camera, M. Fornasa, M. Regis, L. van Waerbeke,J. Harnois-De´raps, S. Ando, M. Bilicki, T. Erben, N. Fornengo, C. Heymans, H. Hildebrandt, H. Hoek-stra, K. Kuijken, and M. Viola, published in Monthly Notices of the Royal Astronomical Society, 467,3, 2706-2722 (2017). The primary authors are T. Tro¨ster, S. Camera, M. Fornasa, M. Regis, and L. vanWaerbeke. T. Tro¨ster led the project, conducted the measurement and analysis, wrote the measure-ment and analysis code, produced all figures, and drafted the manuscript. S. Camera, M. Fornasa, andM. Regis provided the modelling and analytical covariance matrices and contributed to the writing ofSection 2.2. Both the modelling and analytical covariances were validated by comparing it to an inde-pendent code by T. Tro¨ster. L. van Waerbeke (thesis advisor) provided guidance and comments. Allauthors provided comments on the final manuscript.Chapter 3 is based on the article ‘Cross-correlating Planck tSZ with RCSLenS weak lensing: Impli-cations for cosmology and AGN feedback’ by A. Hojjati, T. Tro¨ster, J. Harnois-De´raps, I. G. McCarthy,L. van Waerbeke, A. Choi, T. Erben, C. Heymans, H. Hildebrandt, G. Hinshaw, Y. Z. Ma, L. Miller,M. Viola, and H. Tanimura, accepted for publication in Monthly Notices of the Royal AstronomicalSociety. The primary authors are A. Hojjati, T. Tro¨ster, J. Harnois-De´raps, I. G. McCarthy, L. van Waer-beke. T. Tro¨ster conducted the configuration-space measurements, performed the analysis, wrote themeasurement and analysis code, produced all figures except Figs. 3.1, B.5, and B.4, provided extensivecomments on the manuscript, and addressed the second round of referee comments. Postdoctoral re-searcher A. Hojjati led the project, conducted initial measurements, produced Figs. 3.1, B.5, and B.4,and drafted parts of the manuscript. J. Harnois-De´raps conducted the Fourier-space measurements.I. G. McCarthy provided the simulation products. L. van Waerbeke (thesis advisor) provided the con-ivvergence maps. All authors provided comments on the final manuscript.Chapter 4 is adapted from the article ‘Weak lensing corrections to tSZ-lensing cross correlation’ byT. Tro¨ster and L. van Waerbeke, published in the Journal for Cosmology and Astroparticle Physics, 11,008, 8 (2014), doi:10.1088/1475-7516/2014/11/008. T. Tro¨ster did all calculations, produced the plots,and wrote the manuscript. L. van Waerbeke (thesis advisor) provided guidance and comments.Chapter 5 is based on the article ‘KiDS-450: Tomographic Cross-Correlation of Galaxy Shear withPlanck Lensing’ by J. Harnois-De´raps, T. Tro¨ster, N. E. Chisari, C. Heymans, L. van Waerbeke, M. As-gari, M. Bilicki, A. Choi, H. Hildebrandt, H. Hoekstra, S. Joudaki, K. Kuijken, J. Merten, L. Miller,N. Robertson, P. Schneider, and M. Viola, accepted for publication in Monthly Notices of the RoyalAstronomical Society. The primary authors are J. Harnois-De´raps, T. Tro¨ster, and N. E. Chisari.T. Tro¨ster conducted all measurements and provided comments on the manuscript. Postdoctoral re-searcher J. Harnois-De´raps led the project, conducted the analysis, produced the plots, and drafted themanuscript. N. E. Chisari provided the modelling of intrinsic alignment. All authors provided commentson the final manuscript.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Gravitational lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Heuristic derivation of the lens equation . . . . . . . . . . . . . . . . . . . . . 41.1.2 Derivation of the lens equation based on general relativity . . . . . . . . . . . 51.1.3 Image distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Two-point statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Large-scale structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.1 Halo model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Cross-correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.1 Gamma rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.2 tSZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4.3 CMB lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Cross-correlation of weak lensing and gamma rays: implications for the nature of darkmatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Window functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Three-dimensional power spectrum . . . . . . . . . . . . . . . . . . . . . . . 30vi2.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.1 Weak lensing data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.2 Fermi-LAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.1 Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.4.2 Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.3 Statistical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.5.1 Cross-correlation measurements . . . . . . . . . . . . . . . . . . . . . . . . . 412.5.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Cross-correlating Planck tSZ with RCSLenS weak lensing: implications for cosmologyand AGN feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Observational data and theoretical models . . . . . . . . . . . . . . . . . . . . . . . . 523.2.1 Cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.2 Observational data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.2.3 Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.4 The cosmo-OWLS hydrodynamical simulations . . . . . . . . . . . . . . . . . 583.3 Observed cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.1 Configuration-space analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.2 Fourier-space measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4 Estimation of covariance matrices and significance of detection . . . . . . . . . . . . 623.4.1 Configuration-space covariance . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.2 Fourier-space covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.4.3 Estimating the contribution from the sampling variance . . . . . . . . . . . . . 633.4.4 χ2 analysis and significance of detection . . . . . . . . . . . . . . . . . . . . 643.5 Implications for cosmology and astrophysics . . . . . . . . . . . . . . . . . . . . . . 683.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Weak lensing corrections to tSZ-lensing cross-correlation . . . . . . . . . . . . . . . . . 744.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3.1 Born approximation and lens-lens coupling . . . . . . . . . . . . . . . . . . . 784.3.2 Reduced shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3.3 Redshift distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.3.4 Vector modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86vii4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 KiDS-450: tomographic cross-correlation of galaxy shear with Planck lensing . . . . . . 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3 The data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.3.1 KiDS-450 lensing data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.3.2 Planck κCMB maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.4 The measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4.1 The ξ κCMBγt estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4.2 The CκCMBκgal` estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.4.3 Covariance estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.5 Cosmological inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.5.1 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.5.2 Null tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5.3 Effect of intrinsic alignments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.5.4 Effect of n(z) errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.5.5 Baryon feedback, massive neutrinos and non-linear modelling . . . . . . . . . 1095.5.6 Cosmology from broad n(z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.5.7 Application: photo-z and m-calibration . . . . . . . . . . . . . . . . . . . . . 1125.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.1 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A Supplementary material to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.1 Fourier-space estimator performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 142B Supplementary material to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146B.1 Extra considerations in κ-map reconstruction . . . . . . . . . . . . . . . . . . . . . . 146B.2 Null tests and other effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147C Supplementary material to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152C.1 Fourier space identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152C.2 Convergence - shear relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153C.3 Induced rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155viiiList of TablesTable 2.1 Lensing survey statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Table 2.2 χ20 values of gamma-ray and lensing cross-spectrum measurements. . . . . . . . . . 43Table 3.1 Sub-grid physics of the baryon feedback models in the cosmo-OWLS runs. . . . . . 58Table 3.2 χ2null values before and after including the sampling variance contribution. . . . . . 65Table 3.3 Summary of the statistical analysis of the cross-correlation measurements. . . . . . 67Table 3.4 Summary of χ2min analysis of the cross-correlation measurements from hydrodynam-ical simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Table 5.1 Summary of the KiDS data properties in the different tomographic bins. . . . . . . 94Table 5.2 Summary of χ2, SNR and p-values. . . . . . . . . . . . . . . . . . . . . . . . . . . 102Table 5.3 p-values for the EB test obtained for the six tomographic bins. . . . . . . . . . . . . 104ixList of FiguresFigure 1.1 Geometry of a lensing system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Illustration of geodesic deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.3 Lensing-induced image distortions. . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 1.4 Halo-model correlation function and power spectrum . . . . . . . . . . . . . . . . 16Figure 2.1 Gamma-ray and lensing window functions. . . . . . . . . . . . . . . . . . . . . . 26Figure 2.2 Dark matter clumping factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.3 Gamma-ray intensities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.4 Map of the gamma-ray sky. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.5 Model for angular cross-power spectrum between gamma rays and lensing. . . . . 35Figure 2.6 Measurement of the cross-spectrum between gamma rays and lensing for differentgamma-ray data preparation choices. . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.7 Covariances for gamma-ray and lensing cross-correlation. . . . . . . . . . . . . . 38Figure 2.8 Measurement of the cross-spectrum between gamma rays and weak lensing datafrom CFHTLenS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 2.9 Measurement of the cross-spectrum between gamma rays and weak lensing datafrom RCSLenS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 2.10 Measurement of the cross-spectrum between gamma rays and weak lensing datafrom KiDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 2.11 Measurement and model of cross-spectrum between gamma rays and lensing. . . . 44Figure 2.12 Exclusion limits on the annihilation cross-section 〈σannv〉 and WIMP mass mDMfrom KiDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.13 Exclusion limits on the decay rate Γdec and WIMP mass mDM from KiDS. . . . . . 46Figure 2.14 Exclusion limits on the annihilation cross-section 〈σannv〉 and WIMP mass mDM at2σ significance for CFHTLenS, RCSLenS, and KiDS. . . . . . . . . . . . . . . . 47Figure 2.15 Exclusion limits on the annihilation cross-section 〈σannv〉 and WIMP mass mDM at2σ significance for the combination of CFHTLenS, RCSLenS, and KiDS. . . . . . 48Figure 2.16 Exclusion limits on the decay rate Γdec and WIMP mass mDM at 2σ significance forthe combination of CFHTLenS, RCSLenS, and KiDS. . . . . . . . . . . . . . . . 48Figure 3.1 Redshift distribution, n(z), of the RCSLenS sources for different r-magnitude cuts. 55xFigure 3.2 Cross-correlation measurements of y–κ and y–γt from RCSLenS. . . . . . . . . . 60Figure 3.3 Similar to Fig. 3.2 but for Fourier-space estimator, Cy−κ` . . . . . . . . . . . . . . . 61Figure 3.4 Correlation-coefficient matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.5 Ratios of the variance between the 14 RCSLenS fields and the variance estimatedfrom random shear maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 3.6 SNR as a function of the maximum angular separation. . . . . . . . . . . . . . . . 68Figure 3.7 Comparisons of the cross-correlation measurement from RCSLenS to predictionsfrom hydrodynamical simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 3.8 Same as Fig. 3.7 for the Fourier-space estimator, Cy−κ` . . . . . . . . . . . . . . . 70Figure 4.1 The different contributions to the angular cross-power spectrum Cyκ` . . . . . . . . . 87Figure 4.2 The third-order contribution to the cross-power spectrum computed. . . . . . . . . 88Figure 5.1 Redshift distribution of the selected KiDS-450 sources in the tomographic bin. . . 93Figure 5.2 Cross-correlation measurement between Planck 2015 κCMB maps and KiDS-450lensing data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 5.3 Tomographic measurement of Afid. . . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 5.4 Fractional effect on the signal when changing the fiducial cosmology. . . . . . . . 103Figure 5.5 Strength of the contamination by intrinsic galaxy alignments for different tomo-graphic bins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure 5.6 Fractional effect on the CκCMBκgal` signal when varying n(z). . . . . . . . . . . . . . 108Figure 5.7 Fractional effect of the AGN baryon feedback and massive neutrinos on the cross-spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 5.8 Constraints on σ8 and Ωm as estimated from the cross-correlation measurement,ignoring potential contamination by intrinsic galaxy alignments . . . . . . . . . . . 111Figure 5.9 Same as Fig. 5.8, but here assuming 10% contamination from IA in the cross-correlation measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Figure 5.10 1σ contour regions on the shear calibration correction δm and the redshift distribu-tion correction δz in the bin zB > 0.9. . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 5.11 1σ contour regions on the shear calibration correction δm and the redshift distribu-tion correction δz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure A.1 Validation of power spectrum estimator. . . . . . . . . . . . . . . . . . . . . . . . 143Figure A.2 Validation of power spectrum estimator at small scales. . . . . . . . . . . . . . . . 144Figure B.1 Impact of different magnitude cuts on the y–κ and y–γt cross-correlation signals. . 147Figure B.2 Impact of varying the smoothing of the convergence maps on the y–κ cross-correlation signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure B.3 The impact of masking point sources in the y map on the y–κ and y–γt cross corre-lation analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148xiFigure B.4 Stacked B-mode residual from the RCSLenS fields represented through the auto-correlation function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure B.5 Summary of the null tests performed on the y–κ and y–γt estimators. . . . . . . . . 150xiiChapter 1IntroductionAccording to the Copernican principle, we do not observe the Universe from any special vantage point.In more technical terms, this means that on sufficiently large scales the Universe is statistically homo-geneous (the same everywhere) and isotropic (the same in all directions). On these large scales thedynamics are expected to be dominated by gravity. The most accurate theory of gravity to date is Ein-stein’s general relativity (GR). Its field equations are notoriously difficult to solve in general. However,as is often the case in physics, imposing symmetries allows one to simplify the field equations to apoint where they become tractable. Specifically, upon imposing homogeneity and isotropy on Einstein’sfield equations of GR, one arrives at a solution for an expanding universe, the Friedmann-Lemaıˆtre-Robertson-Walker (FLRW) metric.The expansion history is determined by the energy content of the Universe, specifically by the abun-dances of non-relativistic matter ΩM, radiation ΩR, and the cosmological constant ΩΛ, and is describedby the Friedmann equation. The abundances change with time; relativistic massive particles in the earlyUniverse behave like radiation but the expansion of the Universe causes them to adiabatically cool downto non-relativistic speeds later on and contribute to the non-relativistic (cold) matter budget. Further-more, while the density of non-relativistic matter decreases proportional to the volume of the expandingspace, the radiation density experiences an additional decrease due to the shifting of the radiation fieldto longer wavelengths – the redshift – such that the density of radiation decays quicker than the densityof matter.At early times, matter was in the form of a relativistic plasma of elementary particles. As theUniverse expanded, the plasma cooled, allowing the formation of baryons and light nuclei. At the endof this process, called big bang nucleosynthesis (BBN), the baryon content Universe consisted of aboutthree quarters hydrogen, one quarter helium, and small amounts of deuterium, tritium, and lithium.When the Universe had cooled enough for electrons to be able to combine with the hydrogen andhelium nuclei to form neutral atoms, the Universe became transparent to the photons. Released fromthe primordial plasma, these photons have since travelled the Universe essentially unimpeded and formtoday’s cosmic microwave background (CMB), an isotropic, thermal radiation bath with a temperatureof 2.73 K.Quantum fluctuations of the quantum fields of the elementary particles sourced inhomogeneities in1the primordial plasma. The inhomogeneities expanded as acoustic waves – the baryon acoustic oscilla-tions (BAO) – in the plasma until recombination, when the Universe became transparent to photons. Theinhomogeneities left an imprint on the free streaming photons, observable as small anisotropies in theCMB. These anisotropies have a characteristic length scale corresponding to the distance the acousticwaves travelled in the primordial plasma.Unlike the pressure-supported inhomogeneities in the primordial plasma, the dark matter inhomo-geneities had been free to collapse under the influence of gravity. Once the baryonic inhomogeneities arefree from the photon-pressure that supported them, they too begin to collapse, aided by the existing po-tential wells of dark matter structures. The characteristic correlation length of the BAO is preserved bythe growth of structure and is observable in the distribution of galaxies in the large-scale structure (LSS)of the Universe. In cosmological recent times, the expansion of the Universe has been accelerating,driven by dark energy. This late time accelerated expansion of the Universe can be observed in cosmo-logical distance measures, such as supernovae (SNe) and BAO, and in the suppression of the growth ofstructure in the Universe.The expansion history of the Universe, the primordial abundances of elements, the temperature andanisotropies of the CMB, and the growth of the LSS and its clustering properties form the core of thestandard model of cosmology, called ΛCDM after the two main contribution to today’s energy contentof the Universe: dark energy (Λ) and cold dark matter (CDM).While ΛCDM has been remarkably successful in describing cosmological observations, it positsthe existence of dark matter and dark energy. To date, there is no satisfactory explanation in termsof fundamental physics for either of the two. The work presented in this thesis seeks to illuminate thenature of dark matter by employing gravitational lensing, the deflection of light by matter, in conjunctionwith other probes. In Chapter 2 the correlation of gamma rays with gravitational lensing is used toconstrain particle models of dark matter. Chapters 3 and 4 investigate the relationship between baryonsand dark matter by probing their distributions through the cross-correlation of gravitational lensing andthe thermal Sunyaev-Zeldovich (tSZ) effect. The evolution of structure in Universe and systematicsinherent to the measurement of gravitational lensing are analysed in Chapter 5.The following sections give an introductory exposure to the some of the tools used in this thesis,namely gravitational lensing, its two-point statistics, and its application to LSS.1.1 Gravitational lensingGravitational lensing describes the deflection of light rays by gravitational potentials. The general rela-tivistic solution for the deflection angle of a light ray by a mass M and impact parameter d is α = 4 GMd(see, e.g., Wald, 1984; Carroll, 2004). Remarkably, this expression differs only by an overall factor of 2from a classical calculation.A gravitational lensing system consists of an observer, a source, and one or more, potentially ex-tended, lenses. A simple system with a single lens plane is shown in Fig. 1.1. The photon path from thesource, S, to the observer, O, is shown in blue. In the absence of a lens, the source would be observed ata angular position ~β on the sky. The deflection of light by the lens instead causes the source to appear2Figure 1.1: Geometry of a lensing system. The observer O is on the left and observes a source Sat an angular position ~θ . The (deflected) light path from the source to the observer is shownin blue while the undeflected path is denoted by the dotted line. In the absence of the lens L,the source would be observed at an angular position ~β .at a position ~θ . If the lens deflects the light ray by a deflection angle ~α , then the geometry of Fig. 1.1suggests that ~θ and ~β are related by~β = ~θ − DLSDS~α , (1.1)where DLS is the distance between the lens and the source and DS is the distance between the observerand the source. Here we have assumed that the deflection angle is small enough that the small angle ap-proximation is applicable. In astrophysical situations, the deflection angle is indeed very small, reachingonly a few arcmin in the case of lensing of the CMB. The small angle approximation is therefore welljustified. For multiple, localised lenses Li, the lens equation becomes~β = ~θ −∑iDLiSDS~αi . (1.2)We will now give a heuristic derivation of this lens equation and then justify the result by sketching theformal derivation based on GR, where we will also present the general lens equation Eq. (1.19).31.1.1 Heuristic derivation of the lens equationIn the following section we will forgo mathematical rigour in favour of heuristic and physical arguments.We parametrise the transverse deviation of a light ray from the undeflected path by ~x(χ), where χdenotes the (comoving) distance from the observer, as illustrated in Fig. 1.1. We assume that all anglesare small such that the deviation is well described by a vector perpendicular to the undeflected path.At the observer, the angle between the deflected and undeflected paths is ~ˆα = ~θ −~β . The deflection isobtained by integrating the equation of motion of the transverse deviation. The deviation ~x(χ) can bewritten as~x(χ) =∫ t(χ)0dt ′~˙x(t ′) = χ~ˆα+∫ t0dt ′∫ t ′0dt ′′~¨x(t ′′)= χ~ˆα+1c2∫ χ0dχ ′∫ χ ′0dχ ′′~¨x(χ ′′) = χ~ˆα+1c2∫ χ0dχ ′(χ−χ ′)~¨x(χ ′) ,(1.3)where we used that ~x(0) = 0 and dχdt = c, c being the speed of light. The transverse acceleration ~¨x canbe expressed in terms of the transverse gradient of the Newtonian potential as~¨x(χ) =−~∇⊥φ(~γ0(χ)+~x(χ)) , (1.4)where ~γ0(χ) denotes the location of the undeflected light ray at a distance χ from the observer and thetransverse gradient ~∇⊥ is taken with respect to the direction of the undeflected ray ~γ0. Equation (1.4)applies for classical, massive particles. In the following, we include an extra factor of 2, anticipatingthat the deflection angle of light in GR is twice that of the classical derivation, such that the accelerationis ~¨x(χ) = −2~∇⊥φ(~γ0(χ) +~x(χ)). The classical derivation essentially ignores the spatial part of themetric, whereas in GR the spatial and temporal components contribute equally to the deflection of amassless particle. Since the deflections are small, evaluating the potential along the unperturbed path,i.e., φ(~γ0(χ)+~x(χ)) ≈ φ(~γ0(χ)), is a good approximation. This is the so called Born-approximationand the effects of dropping it are investigated in Chapter 4 (see also Krause et al., 2010; Bernardeauet al., 2012). At the source S, the deviation ~x(χS) is zero. At a distance χS from the observer wetherefore have0 = χS~θ −χS~β − 2c2∫ χS0dχ ′(χS−χ ′)~∇⊥φ(~γ0(χ ′)) , (1.5)or, rearranging the terms and dividing by χS,~β = ~θ − 2c2∫ χS0dχ ′(χS−χ ′)χS~∇⊥φ(~γ0(χ ′)) . (1.6)If the lenses are well localised, i.e., if their extent along the line-of-sight is small compared to thedistances between lenses, the source, and the observer, then we can break the integral up into smallsegments centred on each lens i:∫ χS0 dχ ≈ ∑i∫ χi+εχi−ε dχ . Since the potential varies on much smallerscales than the factor (χS−χ)χS , we can take this factor out of the integral. We are left with integrals of the4form ∫ χi+εχi−εdχ~∇⊥φ(~γ0(χ)) . (1.7)We will now assume that the potential is sufficiently spherically symmetric at the distance ~d from thecentre of the mass distribution. The potential is then simply φ(d,χ) =−GMr =− GM√d2+(χ−χi)2 . Insertingthe potential into Eq. (1.7) we find∫ χi+εχi−εdχ~∇⊥φ(~γ0(χ)) =−2GM~∇⊥arsinh(εd)= 2GM~dd21√1+ d2ε2. (1.8)The distance ε along the line-of-sight is much larger than the impact parameter d, thus reducing Eq. (1.8)to 2GMc2~dd2 . Equation (1.6) can now be written as~β = θ −∑i(χS−χi)χS4GMc2~did2i. (1.9)Noting that χS− χi = DLiS, χS = DS, and 4GMc2~did2iis the deflection angle ~αi, we have therefore derivedEq. (1.2).1.1.2 Derivation of the lens equation based on general relativityThere are a number of ways to derive the lensing equations in GR, for example through geodesic de-viation equation, Fermat’s theorem, or geodesic congruences. Here we sketch a derivation based ongeodesic deviation. For detailed derivations, see, e.g., Blandford et al. (1991), Seitz et al. (1994), Sachs(1961), Pyne et al. (1996), and reviews by Bartelmann et al. (2001) and Bartelmann (2010).We wish to study the evolution of a bundle of light rays. We choose one ray as our fiducial rayγµ0 (λ ), parameterised by the affine parameter λ . The tangent vector along this ray is given byk˜µ =d γ0(λ )µdλ. (1.10)We want to describe the evolution of the deviation between the fiducial ray γµ0 and some adjacent rayγµ , as illustrated in Fig. 1.2. We choose a two-dimensional screen spanned by the vectors Eµ1 and Eµ2 .The two vectors Eµ1,2 are chosen such that they are perpendicular to observer uµ and the direction of theray k˜µ , i.e., Eµi uµ = Eµi k˜µ = 0. Imposing orthonormality further sets Eµi E jµ = δi j. We can now definea deviation vector Y µ that connects the fiducial ray γµ0 to the adjacent ray γµ asY µ =−ξ 1Eµ1 −ξ 2Eµ2 −ξ 0k˜µ , (1.11)where ξ 1,2 parametrises the transversal components of the deviation and ξ 0 the deviation along the ray.5Figure 1.2: Illustration geodesic deviation. From a bundle of light rays we choose a fiducial rayγµ0 and some adjacent ray γµ . The tangent of γµ0 is given by k˜µ . We can define a two-dimensional screen perpendicular to the fiducial ray and the observer spanned by Eµ1 and Eµ2 .The deviation between γµ0 and γµ projected on this screen is then given by ξ i.The evolution of the transverse components is then given byd2ξ idλ 2=T ξ i , (1.12)where T (λ ) is the optical tidal matrix. This matrix can be written in the formT =(R+ℜ[F ] ℑ[F ]ℑ[F ] R−ℜ[F ]), (1.13)whereR describes the isotropic expansion or contraction of the bundle and depends on the Ricci tensor.Anisotropic shearing of the bundle is described byF , which depends on the Weyl curvature tensor.We are interested in the propagation of light in an expanding Universe described by the FLRWmetric with matter inhomogeneities. If the Newtonian potential is small in the sense that φc2  1, themetric can be perturbed around the FLRW background asds2 =−(1+2φc2)c2dt2+a(t)2(1− 2φc2)(dχ2+ fK(χ)2dΩ2). (1.14)Here a denotes the scale factor and fK depends on the global curvature K and comoving distance χ asfK(χ) =1√Ksin(√Kχ)K > 0χ K = 01√−K sinh(√−Kχ) K < 0. (1.15)6Given the metric, the Ricci and Weyl curvature tensors can be calculated, yielding expressions for Rand F . Expressing the deviations equation Eq. (1.12) in terms of the comoving distance χ and thecomoving bundle dimension xi, we finally find(d2dχ2+K)xi =−2∂iφc2. (1.16)The Green’s function of this differential equation isG(χ,χ ′) =1√Ksin(√K(χ−χ ′))Θ(χ−χ ′) = fK(χ−χ ′)Θ(χ−χ ′) , (1.17)where Θ denotes the Heaviside step function. The initial conditions are given byxi(0) = 0 ,dxidχ(0) = θ i−β i , (1.18)where θ i is again the observed position of the source on the sky and β i the position if no deflection hadoccured. The solution for the evolution of the comoving bundle dimension isxi(χ) = fK(χ)(θ i−β i)− 2c2∫ χ0dχ ′ fK(χ−χ ′)∂ iφ(~γ0(χ ′)+~x(χ ′)) . (1.19)Considering the case of a flat Universe, i.e., K = 0 such that fK(χ) = χ , applying the Born approxima-tion such that φ(~γ0(χ ′)+~x(χ ′))≈ φ(~γ0(χ ′)), and evaluating Eq. (1.19) at χS, we recover Eq. (1.5) fromthe heuristic derivation.1.1.3 Image distortionsThe deflection angle and transverse deviation are not observable, as we do not know the true positionof the source. We instead observe distortions of the source. This distortion is given by the Jacobi mapwhich describes the infinitesimal change across the image as the source position is varied. Taking thederivative of ~β with respect to the image position ~θ we find to first orderAi j =∂βi∂θ j= δi j− 2c2∫ χS0dχ ′fK(χS−χ ′) fK(χ ′)fK(χS)∂i∂ jφ(~γ0(χ ′)) , (1.20)where the coordinate system is chosen such that the spatial derivatives ∂i are along directions perpen-dicular to the direction of the ray ~γ0. In Eq. (1.20) we applied the Born approximation and used the ap-proximation ∂∂θ iφ(~γ0(χ′)) = fK(χ ′)∂iφ(~γ0(χ ′)). The general expression without these approximationsis given by Eq. (4.3). The Jacobi map is usually split into trace and trace-free parts and parametrised bythe convergence κ and the shear components γ1 and γ2:Ai j =(1−κ 00 1−κ)−(γ1 γ2γ2 −γ1). (1.21)7The convergence causes isotropic stretching of the source image, whereas the shear causes anisotropicshearing. The effects of the convergence and shear on a circular source image are demonstrated inFig. 1.3. Since the two partial derivatives in Eq. (1.20) commute, the Jacobi map is symmetric at firstorder. The general expression for the Jacobi map Eq. (4.3) does not have this symmetry, however, andthus allows for a rotational component in addition to the convergence and shear.>0κ γ1 γ2<0Figure 1.3: Lensing-induced distortions of a circular source image. The undistorted image isshown on the left. The three columns on the right show the effects of the convergence κand shear components γ1 and γ2.In the following, we will focus on the convergence, as it can be simply related to the matter dis-tribution in the Universe. It is, however, not a direct observable. Galaxy lensing surveys measure theellipticities of large numbers of galaxies. In this thesis we make use of the currently best availablegalaxy lensing data sets Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS), Red ClusterSequence Lensing Survey (RCSLenS), and Kilo-Degree Survey (KiDS), each of which contains ellip-ticity estimates for millions of galaxies over a combined area of& 1000 deg2. In the absence of intrinsicalignments of these source galaxies, the observed ellipticities serve as an estimator of the ‘reduced shear’gi, defined asgi =γi1−κ . (1.22)In weak lensing, the shear and convergence are small so that the reduced shear can be used as anestimator of the shear. Magnification, as observed, e.g., through excess galaxy number counts, measuresthe determinant of the Jacobi map Eq. (1.21), which is a combination of shear and convergence. In CMBlensing, usually the lensing potential is estimated instead, such as through the quadratic estimator of Huet al. (2002). Both the shear and the lensing potential can be related to the convergence, so that we can8restrict ourselves to the convergence in the following without loss of generality.From Eq. (1.21) we see that the convergence is given by κ = 1− 12 trA . It is therefore proportionalto some (weighted) integral over (∂ 21 + ∂ 22 )φ(~γ0(χ)). If the potential is localised, we replace the two-dimensional Laplacian with the three-dimensional one. This allows us to insert the Poisson equation:~∇2φ =32H20c2ΩMaδ , (1.23)where H0 is the Hubble constant today. The convergence at a position~θ on the sky can now be expressedin terms of the line-of-sight integral over the density contrast asκ(~θ ,χS) =32H20ΩMc2∫ χS0dχ ′fK(χS−χ ′) fK(χ ′)a(χ ′) fK(χS)δ ( fK(χ ′)~θ ,χ ′) , (1.24)where we set the undeflected path to ~γ0(χ ′) = ( fK(χ ′)~θ ,χ ′). The response of lensing to the densitycontrast is encapsulated into a window functionW˜ κ(χ,χS) =32H20ΩMc2fK(χS−χ) fK(χ)a(χ) fK(χS). (1.25)So far we have only worked with a single source at a comoving distance χS. This is a valid assumptionin the case of lensing of the CMB, where the source plane is located at the surface of last scattering atz ≈ 1100. In galaxy lensing, we derive the shear and convergence from many source galaxies instead.If the source galaxies follow a distribution p(χS) then the convergence due to this source sample isκ(~θ) =∫ χH0dχS p(χS)κ(~θ ,χS) . (1.26)This dependence on the source distribution can be absorbed into the window functionW κ(χ) =∫ χHχdχ ′p(χ ′)W˜ κ(χ,χ ′) . (1.27)Using this general window function, the expression Eq. (1.24) for the convergence can then be be writtenasκ(~θ) =∫ χH0dχ ′W κ(χ ′)δ ( fK(χ ′)~θ ,χ ′) . (1.28)1.1.4 Two-point statisticsIn cosmology, we are often interested in the two-point statistics of fields, such as: the angular powerspectrum of CMB temperature and polarisation anisotropies; the correlation function of galaxies; orcosmic shear – the correlation of galaxy shapes. This is motivated by the fact that at early times thematter field is a Gaussian random field whose statistics are completely described by two-point statistics.While the fields evolve linearly, they remain Gaussian, such that two-point statistics remain an excellent9description. Once the fields evolve non-linearly and deviate from Gaussianity, two-point statistics are notsufficient anymore to completely describe the statistical properties of the fields. Even though they do notcapture all information contained within the fields, two-point statistics are still valid statistics, however,and provide useful information about the physical processes we wish to measure. Furthermore, theestimation and modelling of higher-order n-point statistics of cosmological observables is a challengein itself and still a topic of current research. As the Fourier modes of a Gaussian random field areindependent and remain so under linear evolution of LSS, it is often advantageous to work in Fourierspace with the power spectrum instead of the configuration space correlation function.The convergence field is defined on the sphere. We therefore decompose it in terms of sphericalharmonics Ylm asκ(~θ) =∞∑l=0l∑m=−lκlmYlm(~θ) , κlm =∫dΩκ(~θ)Y ∗lm(~θ) . (1.29)The integral is over the whole sphere with dΩ= dφdϑ sinϑ where (φ ,ϑ) correspond to the position ~θon the sphere. If the regions of interest on the sky are small, they can be approximated by tangent planesand it is not necessary to employ spherical harmonics. We follow the more general full-sky treatmentfor now. For a derivation using the flat-sky approximation, see Section 4.2.Expanding the correlation function ξκκ(|~θ −~θ ′|) = 〈κ(~θ)κ(~θ ′)〉 in terms of spherical harmonicswe find〈κ(~θ)κ(~θ ′)〉= ∑lml′m′〈κlmκ∗l′m′〉Ylm(~θ)Y ∗l′m′(~θ ′) =∑lmCκκl Ylm(~θ)Y∗lm(~θ′) , (1.30)where the angular power spectrum Cκκl is defined as 〈κlmκ∗l′m′〉 = δll′δmm′Cκκl . Taking the definition ofthe convergence Eq. (1.28) the correlation function can be expressed as〈κ(~θ)κ(~θ ′)〉=∫dχdχ ′W κ(χ)W κ(χ ′)〈δ (χ~θ ,χ)δ (χ ′~θ ′,χ ′)〉 , (1.31)where we have assumed that the Universe is flat, i.e., fK(χ) = χ . Measurements of the CMBanisotropies and the BAO scale have strongly constrained the curvature of the Universe (Planck Col-laboration XIII, 2016), such that this is a justified assumption. We write the density contrast δ (χ~θ ,χ)in term of its Fourier transformδ (~x,χ) =∫ d3k(2pi)3ei~k~xδˆ (~k,χ) , (1.32)where we denoted the three-dimensional position by ~x. Note that χ sets the time at which the densitycontrast is evaluated at position~x. The power spectrum of the density contrast is defined as〈δˆ (~k,χ)δˆ ∗(~k′,χ ′)〉= (2pi)3δD(~k−~k′)Pδδ (|~k|,χ,χ ′) . (1.33)The correlation function of the density contrast 〈δ (χ~θ ,χ)δ (χ ′~θ ′,χ ′)〉 is then related to the density10contrast power spectrum Pδδ (|~k|,χ,χ ′) by〈δ (~x,χ)δ (~x′,χ ′)〉=∫ d3k(2pi)3ei~k(~x−~x′)Pδδ (|~k|,χ,χ ′) . (1.34)The Fourier basis ei~k~x can be expanded in terms of spherical Bessel functions jl and spherical harmonicsYlm asei~k~x = 4pi∑lmil jl(kχ)Y ∗lm(~θ)Ylm(~ω) , (1.35)where ~θ is the angular direction of vector~x and ~ω the angular direction of the wave vector~k. Insertingthis expansion into Eq. (1.34), we find that〈δ (~x,χ)δ (~x′,χ ′)〉= 2pi∑lm∫dkk2 jl(kχ) jl(kχ ′)Y ∗lm(~θ)Ylm(~θ′)Pδδ (k,χ,χ ′) , (1.36)where we used the orthogonality relation of the spherical harmonics∫dΩYlm(~θ)Y ∗l′m′(~θ) = δll′δmm′ . (1.37)The spherical Bessel functions in Eq. (1.34) oscillate with a period of ∼ 2piχ as a function of k. For cos-mological distances χ , this causes the Bessel functions to vary rapidly compared to the power spectrumPδδ (k). The power spectrum can then be approximated by evaluating it at the peak of the sphericalBessel function k ∼ lχ and taking it out of the integral. Applying the orthogonality relation of thespherical Bessel function we thus arrive at∫ ∞0dkk2 jl(kχ) jl(kχ ′)Pδδ (k,χ,χ ′)≈pi2χ2δD(χ−χ ′)Pδδ(lχ,χ). (1.38)This is the so called Limber approximation (Limber, 1953), which is sufficiently accurate for all currentpractical applications. The deviations from the exact treatment are less than 1% for l& 50. For a detailedanalysis of the effects of the Limber and other approximations on the convergence power spectrum, seeKilbinger et al. (2017). The double line-of-sight integral in Eq (1.31) can now be reduced to one.Comparing to Eq. (1.31) we finally arrive at the angular power spectrum of the convergence:Cκκl ≈∫ χH0dχW κ(χ)2χ2Pδδ(lχ,χ). (1.39)This result can be easily generalised to any field that can be written in terms of a line-of-sight inte-gral. For two fields on the sphere A and B, sourced by three-dimensional fields A˜ and B˜, with windowfunctions W A and W B, the angular (cross) power spectrum readsCABl ≈∫ χH0dχW A(χ)W B(χ)χ2PA˜B˜(lχ,χ). (1.40)11All probes studied in this thesis are of this form, allowing a single framework to handle cross-correlations between weak lensing, gamma rays, tSZ, and CMB lensing.While power spectra are more natural from a theoretical perspective, it is often easier to measurecorrelation functions in configuration space, since estimators for correlation functions consist of simplesums over products of the quantities that are being correlated, see, e.g, Eq. (2.11). Power spectrumestimators are more complicated: the estimator in Chapter 2 relies on an integral over the measuredcorrelation function while the pseudo-C` estimator in Chapter 3 explicitly needs to account for masks.In this thesis we consider two types of correlation functions: two-point correlation functions ξ κXbetween the convergence κ and some other scalar field X , such as the tSZ y parameter or CMB lensingconvergence; and the two-point correlation function ξ γt X between the tangential shear γt and a scalarfield X . The tangential shear γt describes the tangential component of the shear with respect to somereference point (see, e.g, Viola et al., 2015, for a technical definition). The tangential shear two-pointcorrelation function is estimated by measuring the tangential component of the source galaxy elliptici-ties around all pixel positions of the map of the scalar field X . The tangential shear two-point correlationfunction ξ γt X has the advantage that it operates directly on the raw data sets – the galaxy shape cata-logues, such as those from CFHTLenS, RCSLenS, or KiDS and scalar maps, such as those provided bythe Planck team – and thus does not require additional processing steps that could introduce systematics.The correlation functions can be related to the power spectrum Eq. (1.40) byξ κX(ϑ) =12pi∫ ∞0d``J0(`ϑ)CκX` and ξγt X(ϑ) =12pi∫ ∞0d``J2(`ϑ)CκX` , (1.41)where ϑ denotes the angular separation on the sky and J0 and J2 are the Bessel functions of the first kindof order zero and two, respectively. Note that both ξ κX and ξ γt X are related to the angular cross-powerspectrum CκX` , such that the underlying modelling is the same for the two estimators.1.2 Large-scale structureThe convergence power spectrum Eq. (1.39) probes the matter power spectrum at different epochs. Inthe linear regime, the time evolution of the Fourier coefficients of the matter field can be separated intoa growth factor D+(t) and the Fourier coefficients of the linear density contrast at some initial time asδˆ lin(~k, t) = D+(t)δˆ lin0 (~k) . (1.42)The differential equation governing the growth of structure has two solutions, a growing mode D+ anda decaying mode D−. Since we are interested in the matter distribution at late times, the decaying modecan be neglected in our discussion. Because the time evolution separates from the spatial dependence,the time evolution of the linear power spectrum is therefore simplyPlinδδ (k, t) = D+(t)2Plinδ0δ0(k) . (1.43)12The linear evolution breaks down at scales where ∆2,lin(k, t) = k34pi2 Plinδδ (k, t) & 1. At small scales ∆2,linhas a ∼ logk2 dependence, such that small scales become non-linear before large scales. These scalesare then subject to non-linear gravitational collapse. Describing the non-linear evolution of the matterfield has proven difficult. Extending the perturbative treatment to higher orders allows us to describe thepower spectrum accurately into the mildly non-linear regime. However, to model the power spectrumdown to small, truly non-linear scales, it is necessary to rely on N-body simulations or phenomenologicalmodels, such as the halo model.1.2.1 Halo modelIn the halo model it is assumed that all matter has collapsed into haloes. The matter distribution canthen be written as (see, e.g., Scherrer et al., 1991; Scoccimarro et al., 2001)ρ(~x) =∑iρi(~x−~xi) , (1.44)where the sum runs over all haloes i, and ρi is the density profile of halo i with centre ~xi. It has beenshown in N-body simulations that dark matter haloes on average are well described by a sphericallysymmetric profile ρh(r|M,z) that only depends on the mass and redshift (Navarro et al., 1997). Thedensity profiles in Eq. 1.44 can thus be written as ρh(~x−~xi|Mi,z) = ρh(|~x−~xi||Mi,z). In the following,we will investigate statistics of the density field. In order to calculate the expectation values, it is usefulto separate the mass and position dependence of the haloes from their profile:ρ(~x) =∫dMd3y∑iδD(M−Mi)δD(~y−~xi)ρh(|~x−~y||M,z) . (1.45)The expectation value of the density field can then be expressed as〈ρ〉=∫dMd3y 〈∑iδD(M−Mi)δD(~y−~xi)〉ρh(|~x−~y||M,z) . (1.46)The term in the brackets describes the mass and position of haloes and has dimensions of M−1L−3(inverse mass times inverse length cubed). The expectation value of this term should therefore be anumber density per mass. We hence set〈∑iδD(M−Mi)δD(~y−~xi)〉= dn(M,z)dM , (1.47)where dn(M,z)dM is the mass function, which describes the number density of haloes at redshift z withmasses between M and M+dM. Equation (1.46) can now be simplified to〈ρ〉=∫dMdn(M,z)dM∫d3y ρh(|~x−~y||M,z) = ρ¯ , (1.48)13such that we indeed recover the average density of the Universe ρ¯ . In the second equality we have usedthat the spatial integral over the density profile is just the mass M of the halo and that the mass integralover M dn(M,z)dM is the average matter density.The quantities of interest in this thesis are two-point statistics, such as two-point correlation func-tions and power spectra. The two-point correlation function of the density field is ξρρ(~x,~x−~r) =〈ρ(~x)ρ(~x−~r)〉. Since the Universe is assumed to be statistically homogeneous, the correlation functiononly depends on the separation: ξρρ(~x,~x−~r) = ξρρ(~r). Furthermore, isotropy demands that the corre-lation function only depends on the modulus of the separation, not its direction, i.e., ξρρ(~r) = ξρρ(|~r|).Using the formalism from Eq. (1.44), the correlation function ξρρ(|~r|) can be separated into two terms:ξρρ(|~r|) = 〈ρ(~x)ρ(~x−~r)〉= 〈∑iρh(|~x−~xi||Mi,z)ρh(|~x−~r−~xi||Mi,z)〉+ 〈∑i 6= jρh(|~x−~xi||Mi,z)ρh(|~x−~r−~x j||M j,z)〉 .(1.49)The first term describes the correlation within haloes and is called the ‘one-halo’ term. The second term,called the ‘two-halo’ term, describes the correlation between different haloes.Following the same steps as in the derivation of the mean density in Eq. (1.48), we can write theone-halo term as〈ρ(~x)ρ(~x−~r)〉1h =∫dMdn(M,z)dM∫d3yρh(|~x−~y||M,z)ρh(|~x−~r−~y||M,z) . (1.50)The two-halo term can be treated analogously:〈ρ(~x)ρ(~x−~r)〉2h =∫dM1dM2∫d3y1d3y2×〈∑i6= jδD(M1−Mi)δD(M2−M j)δD(~y1−~xi)δD(~y2−~x j)〉×ρh(|~x−~y1||M1,z)ρh(|~x−~r−~y2||M2,z) .(1.51)The interpretation of the term in the second line is similar to that of Eq. (1.47): it should describe thenumber densities of haloes of masses M1 and M2. Furthermore, since it deals with separate haloes, itshould account for the correlation between them. We therefore define〈∑i 6= jδD(M1−Mi)δD(M2−M j)δD(~y1−~xi)δD(~y2−~x j)〉=dn(M1,z)dMdn(M2,z)dMξhh(|~y1−~y2|,M1,M2) , (1.52)where ξhh(|~y1−~y2|,M1,M2) is the halo correlation function of haloes of masses M1 and M2, separatedby |~y1−~y2|. Putting things together, we are left with the expression for the two-halo contribution to the14matter density correlation function:〈ρ(~x)ρ(~x−~r)〉2h =∫dM1dM2dn(M1,z)dMdn(M2,z)dM∫d3y1d3y2×ρh(|~x−~y1||M1,z)ρh(|~x−~r−~y2||M2,z)ξhh(|~y1−~y2|,M1,M2) .(1.53)The integrals over R3 in Eq. (1.50) and Eq. (1.53) are convolutions and it is therefore advantageousto proceed in Fourier space, where they can be represented as simple products (see Appendix C.1). TheFourier transform of the correlation function is the power spectrum and is defined as〈ρˆ(~k)ρˆ(~k′)〉= (2pi)3δD(~k−~k′)Pρρ(|~k|) , (1.54)where we have again assumed statistical homogeneity and isotropy. The one-halo and two-halo termsof the matter density power spectrum are thenP1hρρ(k,z) =∫dMdn(M,z)dM|ρˆ(k,M,z)|2 ,P2hρρ(k,z) =∫dM1dM2dn(M1,z)dMdn(M2,z)dMρˆ(k,M1,z)ρˆ(k,M2,z)Phh(k,M1,M2,z) .(1.55)The correlation function and power spectrum of the density contrast are shown in Fig. 1.4, demon-strating the scale dependence of one-halo and two-halo terms. Except at the transition scale of around2 Mpc h−1, the correlation is clearly dominated by either the intra-halo correlation, described by theone-halo term, or the inter-halo correlation, described by the two-halo term. The separation into the twocontributions is thus sensible.It is conventional in cosmology to work with the matter density contrast δ (~x) = ρ(~x)−ρ¯ρ¯ instead ofthe matter density itself. The correlation function of the density contrast ξδδ is related to the correlationfunction of the matter density by ξδδ = 1ρ¯2 ξρρ−1. The density contrast power spectrum Pδδ in terms ofthe matter density power spectrum is therefore given by Pδδ = 1ρ¯2 Pρρ − (2pi)3δD(0). The second termonly applies for k = 0, which is not observable, since it corresponds to infinitely large scales. We willthus not include it explicitly in the rest of this discussion.The halo power spectrum Phh(k,M1,M2,z) describes the clustering of haloes, i.e., it is the powerspectrum of the halo number density contrast δ h. The halo number density is biased with respect tothe matter distribution. At large scales, where the two-halo term dominates, the bias is linear andscale-independent such that we can write δˆ h(~k,M,z) = b(M,z)δˆ (~k,z) (Mo et al., 1996; Sheth et al.,1999). It follows that the halo power spectrum can be expressed in terms of the matter power spectrumas Phh(k,M1,M2,z) ≈ b(M1,z)b(M2,z)Pδδ (k,z). At intermediary and small scales this overestimatesthe halo power spectrum, however (Cooray et al., 2002). To resolve this overestimation, a commonapproximation is to estimate the halo power spectrum asPhh(k,M1,M2,z)≈ b(M1,z)b(M2,z)Plinδδ (k,z) , (1.56)1510−2 10−1 100 101r [Mpc h−1]10−1110−1010−910−810−710−6ξ δδ(r)1h2htotallinear10−2 10−1 100 101 102k [Mpc−1 h]10−1210−1110−1010−910−810−710−610−5Pδδ(k)[Mpc3h−3]1h2htotallinearFigure 1.4: Left: Two-point correlation function of the density contrast. The one-halo term isshown in red, the two-halo term in blue, the total in purple, and the prediction from lineartheory as black dotted line. On small scales, the one-halo term dominates, while at scaleslarger than a few Mpc the two-halo term determines the correlation. At these large scales thecorrelation is well described by linear theory. Right: Power spectrum of the density contrast.The formatting is analogous to that of the correlation function.where Plinδδ is the linear matter (density contrast) power spectrum. This allows us to separate the twointegrals of the two-halo term in Eq. (1.55). Since the bias parameter obeys the consistency relation∫dM Mdn(M,z)dMb(M,z) = ρ¯ , (1.57)the two-halo terms takes an especially simple form at large scales: P2hδδ ≈ Plinδδ . This can be clearly seenin Fig. 1.4, where the two-halo term and linear prediction match up to scales within the one-halo regime.In the literature, the halo density profile ρh(r) is often written in terms of a normalised profileu(r,M) =ρ(r,M)∫d3rρ(r,M)=ρ(r,M)M. (1.58)The Fourier transform of the normalised profile uˆ(k,M) then has the property that uˆ(k,M)→ 1 for k→ 0.The one-halo term therefore goes to a constant as k→ 0, as shown in Fig. 1.4. Writing the expressionsin Eq. (1.55) in terms of the density contrast and normalised halo profile, as well as approximating thehalo power spectrum by the linear power spectrum times the bias parameter, we recover the standardresults from the literature:P1hδδ (k,z) =1ρ¯2∫dM M2dn(M,z)dM|uˆ(k,M,z)|2 ;P2hδδ (k,z) =1ρ¯2(∫dM Mdn(M,z)dMb(M,z)uˆ(k,M,z))2Plinδδ (k,z) .(1.59)16Halo model for cross-correlationsThe formalism presented in the previous section can be straightforwardly generalised to describe thecross-correlation between two different fields. For example, the cross-correlation between gravitationallensing and gamma rays in Chapter 2 seeks to constrain the contribution of weakly interacting massiveparticle (WIMP) dark matter annihilation to the extragalactic gamma-ray flux. The gamma-ray flux dueto dark matter annihilations is proportional to the dark matter density squared, since each annihilationevent requires two dark matter particles. Gravitational lensing depends on the density contrast instead(cf. Section 1.1). We therefore wish to model the quantity 〈ρ2DMδ 〉. This can be accomplished by justreplacing one factor of the halo density profile in Eq. (1.55) with the square of the density profile. Themass function and halo power spectrum do not have to be changed since the square of the density isstill described by the same halo distribution. The resulting cross-power spectrum between the densitycontrast and density contrast squared is given by Eq. (2.9), although note the different definition of uthere.The cross-correlation between gravitational lensing and the tSZ effect can be treated similarly: hereone factor of the density is instead replaced by the electron pressure profile. The resulting (angular)cross-power spectrum Eq. (3.14) is therefore only a minor modification of that derived in the previoussection. This versatility is one of the great advantages of the halo model formalism.It should be noted that the halo model is not without its downsides. It is a purely phenomenologicalmodel, making it difficult to extract physical information. Moreover, while it is quite accurate at largeand small scales, this is not the case in the transition regime between the one and two-halo term. Therehave been efforts to ameliorate these inaccuracies by replacing certain terms in the halo model descrip-tion with fitting functions tuned to N-body simulations (see, e.g., Smith et al., 2003; Takahashi et al.,2012; Mead et al., 2015, 2016). These tuned halo models remain computationally very simple, makingit possible to include them in likelihoods sampled by Markov chain Monte Carlo (MCMC) methods,while retaining most of the information content of the simulations on two-points statistics. This com-putational simplicity is especially beneficial in cases where cosmological hydrodynamical simulationsare required, such as for modelling the tSZ effect, since these simulations are still very computationallyexpensive to run.1.3 Cross-correlationsThis thesis investigates the correlations between weak gravitational lensing and other cosmologicalprobes. Specifically, we are interested in two-point statistics, such as correlation functions and powerspectra. In general, we measure two fields Aˆ and Bˆ, which consist of the physical signals of interest A,B and noise components nA, nB:Aˆ = A+nA ; Bˆ = B+nB . (1.60)The noise components encompass any contributions to the measurements that are not the signal ofinterest, e.g., measurement noise or systematics. In the cases we are considering in this thesis, it is17assumed that the noise components do not correlate with the primary signals, i.e., 〈AnA〉 = 〈AnB〉 =〈BnB〉 = 〈BnA〉 = 0, nor do the noise components of the different fields correlate: 〈nAnB〉 = 0. Forexample, in the cross-correlation between gravitational lensing and gamma rays in Chapter 2, the noisecomponent of gravitational lensing comes from the uncertainty in the measurement of galaxy shapes,while the noise component of the extra-galactic gamma ray flux consists of shot-noise due to the finiteobserved photon counts and contamination due to Galactic gamma-ray emissions. Gravitational lensingprobes the extra-galactic matter distribution, while the shape measurement uncertainties are dominatedby the effects of the atmosphere, telescope, and detector on the observed galaxy image. It is thereforewell justified to assume that these two effect are uncorrelated. Similarly, shot-noise of the gamma-rayobservations and Galactic foregrounds are neither expected to correlate with the extra-galactic nature ofgravitational lensing nor with shape measurement effects.The cross-correlation between the two measured fields is therefore given by〈AˆBˆ〉= 〈AB〉 , (1.61)i.e., the measured cross-correlation is an unbiased estimator of the cross-correlation of the true sig-nals. This is in contrast to the auto-correlation 〈AˆAˆ〉 = 〈AA〉+ 〈nAnA〉, which is biased by the noiseauto-correlation 〈nAnA〉. The absence of this noise bias is an important advantage of cross-correlationsover auto-correlations, since the noise auto-correlation often dominates the true auto-correlation signal,sometimes by orders of magnitude, and can be difficult to remove accurately.Even though the cross-correlation measurement is unbiased by the noise, the noise auto-correlationsstill contributes to the error budget and need to be accounted for there. To demonstrate this, we considerthe angular power spectrum CAB` between the two two-dimensional fields A and B defined on the wholesphere. The estimator for CAB` isCˆAB` =12`+1`∑m=−`Aˆ`mBˆ∗`m , (1.62)where Aˆ`m and Bˆ`m are the spherical harmonic coefficients of the observed fields Aˆ and Bˆ. The covarianceof the measured angular cross-power spectrum CˆAB` is thenCov[CˆAB` ,CˆAB`′]= 〈CˆAB` CˆAB`′ 〉−〈CˆAB` 〉〈CˆAB`′ 〉 . (1.63)Inserting the definition of the estimator Eq. (1.62) into the first term we find〈CˆAB` CˆAB`′ 〉=1(2`+1)(2`′+1) ∑mm′〈Aˆ`mBˆ∗`mAˆ`′m′Bˆ∗`′m′〉=1(2`+1)(2`′+1) ∑mm′(〈Aˆ`mBˆ∗`mAˆ`′m′Bˆ∗`′m′〉c+ 〈Aˆ`mBˆ∗`m〉〈Aˆ`′m′Bˆ∗`′m′〉+〈Aˆ`mAˆ`′m′〉〈Bˆ∗`mBˆ∗`′m′〉+ 〈Aˆ`mBˆ∗`′m′〉〈Aˆ`mBˆ∗`′m′〉).(1.64)18Using the definition of the angular power spectrum 〈A`mB∗`′m′〉 = δ``′δmm′CAB` and noting that the thesecond term on the second line in Eq. (1.64) cancels the second term in Eq. (1.63), we can write thecovariance of CˆAB` asCov[CˆAB` ,CˆAB`′]=δ``′2`+1(CˆAA` CˆBB` +(CˆAB`)2)+ Tˆ AB``′ , (1.65)with the trispectrum term Tˆ AB``′ defined asTˆ AB``′ =1(2`+1)(2`′+1) ∑mm′〈Aˆ`mBˆ∗`mAˆ`′m′Bˆ∗`′m′〉c . (1.66)We now split the observed fields again into the true signal and noise component such that we can writeCˆAA` = CAA` +NA` , where CAA` is the angular power spectrum of the true signal A and NA` is the autospectrum of the noise nA. Since the noise terms nA and nB do not correlate with the signals nor witheach other, we again have CˆAB` =CAB` and TˆAB``′ = TAB``′ . The covariance in terms of the true signal powerspectra and noise spectra isCov[CˆAB` ,CˆAB`′]=δ``′2`+1(CAA` CBB` +(CAB`)2+CAA` NB` +CBB` NA` +NA` NB`)+T AB``′ . (1.67)The errors and covariance of the cross-correlation CˆAB` therefore do not only depend on the noise spectraNA` and NB` but also on the auto spectra CAA` and CBB` , the cross-spectrum CAB` , and the trispectrum TAB``′ .The terms in the covariance that depend on the true signals A and B constitute the sampling variance.The sampling variance describes the uncertainty of our measurements due to the finite volume we haveobserved.Broadly speaking, there are three approaches to estimate the covariance in practice: modelling ofthe covariance according to Eq. (1.67); creating many realisations of the data from simulations andestimating the covariance between these realisations; and internal estimates such as bootstrap (Efron,1979) and jackknife (Tukey, 1958) that estimate the covariance by resampling the data. The threeapproaches suffer from different shortcomings. Modelling the covariance can be difficult, especiallywhen contributions from the sampling variance are non-negligible, since the covariance depends on thesignal we wish to measure and for which we might not have an adequate model. Using simulationsto create realisations of the data can become very computationally expensive if the signal is producedby non-linear processes. Finally, covariances of correlated data estimated using bootstrap or jackknifemethods can be biased and need to be checked and calibrated against other covariance estimates.1.4 Thesis overviewThe following chapters of this thesis concern themselves with various aspects of probing the natureof dark matter with gravitational lensing. Specifically, they cover the cross-correlation of gravitationallensing with three other probes: gamma rays; the tSZ effect; and CMB lensing.191.4.1 Gamma raysThe analysis presented in Chapter 2 attempts to answer the question of the microscopic nature of darkmatter. A popular hypothesis for the nature of dark matter is a WIMP. Such a particle would have beenin thermal equilibrium with standard model matter at early times. An attractive feature of the WIMPscenario is that an annihilation cross-section at the weak scale yields the correct dark matter abundanceΩDM today. This annihilation cross-section is about 〈σannv〉∼ 3×10−26 cm3s−1 and is called the thermalrelic cross-section (see, e.g., Jungman et al., 1996).The study of the cross-correlation between gravitational lensing and gamma rays in Chapter 2 ismotivated by the rather generic prediction that the annihilation or decay products of WIMP dark matterwill include gamma rays (Bergstro¨m et al., 2001; Ullio et al., 2002). If dark matter is a WIMP, thenregions of the Universe with high dark matter content are expected to show excesses in gamma raysemission. Gravitational lensing probes the matter distribution in the Universe and, since the mattercontent is dominated by dark matter, it is also a good probe of the dark matter distribution. A correlationbetween gamma rays and gravitational lensing would therefore support the hypothesis that dark matter isa WIMP with annihilation or decay channels into gamma rays. Conversely, a lack of correlation wouldconstrain the parameter space of WIMP dark matter, restricting the allowed masses, annihilation cross-sections, and decay rates. This reasoning is complicated by the fact that there are known astrophysicalgamma-ray sources in the Universe that also trace the matter distribution (Xia et al., 2015). Furthermore,the gamma-ray flux due to dark matter annihilations is strongly dependent on the small-scale clusteringof dark matter. We model different populations of astrophysical gamma-ray emitters and consider arange of small-scale dark matter clustering models to address these uncertainties.1.4.2 tSZMost of matter in the Universe, 84% (Planck Collaboration XIII, 2016), is dark matter. The remaining16% is in the form of baryons, which closely trace the dark matter distribution at large scales but candeviate strongly at small scales where baryon physics become important. Unlike dark matter, baryonsare able to cool radiatively, allowing them to form much denser structures, such as stars and galaxies.These cooling processes increase the clustering of matter at small scales. Other processes, such asstellar winds, super novea, and active galactic nuclei (AGN) feedback cause gas to be redistributed andpossibly expelled from its host galaxy (van Daalen et al., 2011). These processes are highly non-linear,making their modelling complicated. The physical processes typically originate on scales much smallerthan the resolution of cosmological simulations, thus requiring separate sub-grid modelling to includethese astrophysical processes. It is therefore necessary to provide observational data to calibrate thesesimulations.In Chapter 3 we provide such data by comparing the distribution of gas inferred from the tSZ effectto that of the total matter distribution measured by gravitational lensing. The tSZ effect is caused bythe inverse Compton scattering of CMB photons off hot electrons (Sunyaev et al., 1972). The resultingspectral distortion of the CMB can be imaged, yielding a map of the distribution of hot, free electronswhich traces the diffuse gas in the Universe. The tSZ effect is therefore ideal to statistically measure the20effect of violent processes, such as AGN feedback, on gas.The first-order derivation of the convergence power spectrum in Eq. (1.39) has been shown to besufficiently accurate for current and upcoming surveys. However, it is a priori not evident that the sameis true for cross-correlations between the convergence and other fields. To ensure that models used inChapter 3 are not biased by neglecting terms of higher order in the potential, we check in Chapter 4 theeffect of terms that are third and fourth order in the gravitational potential on the cross-power spectrumbetween lensing and tSZ.1.4.3 CMB lensingThe analyses in Chapters 2 and 3 made the implicit assumption that the growth of structure proceeds aspredicted by the ΛCDM concordance model of cosmology and that the measurement of weak gravita-tional lensing are accurate and not biased by systematics. Chapter 5 tests these assumptions by combin-ing two different realisations of gravitational lensing: galaxy lensing; and lensing of the CMB. Galaxylensing from surveys such as CFHTLenS, RCSLenS, and KiDS measures the ellipticities of galaxies toobtain an estimate of the shear field. This kind of lensing was employed in the analyses of Chapters 2and 3. If the galaxy sample has redshift estimates, we can in principle select galaxies to follow a sourcedistribution p(χS) of our choosing. This allows us to optimise the window function W κ in Eq. (1.27)to be sensitive to a certain redshift range. Using different galaxy selections to probe different redshiftranges, a technique called tomography, we have a tool to probe the evolution of the matter distribution.Lensing of the CMB manifests itself in coherent distortions of the CMB anisotropies. By analysingthese distortions we are able to directly estimate the lensing potential. Since the source plane for CMBlensing is the surface of last scattering, it probes the matter distribution of the whole observable Uni-verse. However, since there is only one source plane, it is not possible to do tomographic measurementswith just CMB lensing alone. Cross-correlating galaxy lensing with CMB lensing in redshift slicesallows for a tomographic measurements of the matter distribution while enjoying the advantages ofcross-correlations laid out in Section 1.3. Chapter 5 presents such a cross-correlation and analyses itfor deviations from the expected growth rate of structure and residual systematics in the galaxy lensingdata. The measurements build on previous work presented in Harnois-De´raps et al. (2016), to which theauthor of this thesis provided the measurements, covariances, and analysis of the configuration spaceestimators.21Chapter 2Cross-correlation of weak lensing andgamma rays: implications for the natureof dark matterWe measure the cross-correlation between Fermi-LAT gamma-ray photons and over 1000 deg2 of weaklensing data from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS), the Red ClusterSequence Lensing Survey (RCSLenS), and the Kilo-Degree Survey (KiDS). We present the first mea-surement of tomographic weak lensing cross-correlations and the first application of spectral binningto cross-correlations between gamma rays and weak lensing. The measurements are performed usingan angular power spectrum estimator, while the covariance is estimated using an analytical prescrip-tion. We verify the accuracy of our covariance estimate by comparing it to two internal covarianceestimators. Based on the non-detection of a cross-correlation signal, we derive constraints on weaklyinteracting massive particle (WIMP) dark matter. We compute exclusion limits on the dark matter an-nihilation cross-section 〈σannv〉, decay rate Γdec, and particle mass mDM. We find that in the absenceof a cross-correlation signal, tomography does not significantly improve the constraining power of theanalysis compared to a non-tomographic measurement. Assuming a strong contribution to the gamma-ray flux due to small-scale clustering of dark matter and accounting for known astrophysical sources ofgamma rays, we exclude the thermal relic cross-section for particle masses of mDM . 20 GeV. Theseconstraints are comparable to those derived from other analyses of the extra-galactic gamma-ray fluxbut somewhat weaker than those from local probes, such as from dwarf spheroidal galaxies (dSphs) andthe Galactic centre.2.1 IntroductionThe matter content of the Universe is dominated by so called dark matter, whose cosmological abun-dance and large-scale clustering properties have been measured to high precision (e.g. Hinshaw et al.,2013; Planck Collaboration XIII, 2016; Ross et al., 2015; Anderson et al., 2014; Mantz et al., 2015;Hoekstra et al., 2015; Hildebrandt et al., 2017). However, little is known about its microscopic nature,22beyond its lack of – or at most weak – non-gravitational interactions with standard model matter.WIMPs thermally produced in the early Universe are among the leading dark matter candidates.With a mass of the order of GeV or TeV, their decoupling from thermal equilibrium occurs in thenon-relativistic regime. The weak interaction rate with lighter standard model particles furthermoreensures that their thermal relic density is naturally of the order of the measured cosmological darkmatter abundance (Lee et al., 1977; Gunn et al., 1978).Many extensions of the standard model of particle physics predict the existence of new massiveparticles at the weak scale; some of these extra states can indeed be ‘dark’, i.e., be colour and electro-magnetic neutral, with the weak force and gravity as the only relevant coupling to ordinary matter (forreviews, see e.g., Jungman et al., 1996; Bertone et al., 2005; Schmaltz et al., 2005; Hooper et al., 2007;Feng, 2010).The weak coupling allows us to test the hypothesis of WIMP dark matter: supposing that WIMPsare indeed the building blocks of large-scale structure (LSS) in the Universe, there is a small but finiteprobability that WIMPs in dark matter haloes annihilate or decay into detectable particles. These stan-dard model particles produced by these annihilations or decays would manifest as cosmic rays, whichcan be observed. In particular, since the WIMP mass is around the electroweak scale, gamma rays canbe produced, which can be observed with ground-based or space-borne telescopes, e.g., the Large AreaTelescope on the Fermi gamma-ray space telescope (Fermi-LAT, Atwood et al., 2009). Indeed, analy-ses of the gamma-ray sky have already been widely used to put constraints on WIMP dark matter, seee.g. Charles et al. (2016) for a recent review.The currently strongest constraints on the annihilation cross-section and WIMP mass come from theanalysis of local regions with high dark matter content, such as dSphs (Ackermann et al., 2015c). Theseanalyses exclude annihilation cross-sections larger than ∼3×10−26 cm3s−1 for dark matter candidateslighter than 100 GeV. This value for the annihilation cross-section is known as the thermal cross-section,below which many models of new physics predict dark matter candidates that yield a relic dark matterdensity in agreement with cosmological measurements of the dark matter abundance (Jungman et al.,1996).Instead of these local probes of dark matter properties, one could consider the unresolved gamma-raybackground (UGRB), i.e., the cumulative radiation produced by all sources that are not bright enough tobe resolved individually. Correctly modelling the contribution of astrophysical sources, such as blazars,star-forming, and radio galaxies, allows the measurement of the UGRB to be used to constrain thecomponent associated with dark matter (Fornasa et al., 2015). Indeed, the study of the energy spectrumof the UGRB (Ackermann et al., 2015d), as well as of its anisotropies (Ando et al., 2013; Fornasa et al.,2016) and correlation with tracers of LSS (Ando et al., 2014; Shirasaki et al., 2014; Fornengo et al.,2015; Regis et al., 2015; Cuoco et al., 2015; Shirasaki et al., 2015; Ando et al., 2016; Shirasaki et al.,2016; Feng et al., 2017) have yielded independent and competitive constraints on the nature of darkmatter.In this chapter, we focus on the cross-correlation of the UGRB with weak gravitational lensing.Gravitational lensing is an unbiased tracer of matter and thus closely probes the distribution of dark23matter in the Universe. This makes it an ideal probe to cross-correlate with gamma rays to investigatethe particle nature of dark matter (Camera et al., 2013).We extend previous analyses of cross-correlations of gamma rays and weak lensing of Shirasakiet al. (2014, 2016) by adding weak lensing data from the Kilo-Degree Survey (KiDS, de Jong et al.,2013b; Kuijken et al., 2015) and making use of the spectral and redshift information contained withinthe data sets. This chapter presents the first tomographic weak lensing cross-correlation measurementand the first application of spectral binning to the cross-correlation between gamma rays and galaxylensing. Exploiting tomography and the information contained in the energy spectrum of the dark matterannihilation signal has been shown to greatly increase the constraining power compared to the casewhere no binning in redshift or energy is performed (Camera et al., 2015).The structure of this chapter is as follows: in Section 2.2 we introduce the formalism and theory; thedata sets are described in Section 2.3; Section 2.4 introduces the measurement methods and estimators;the results are presented in Section 2.5; and we draw our conclusions in Section FormalismOur theoretical predictions are obtained by computing the angular cross-power spectrum Cgκ` betweenthe lensing convergence κ and gamma-ray emissions for different classes of gamma-ray sources, de-noted by g. In the Limber approximation (Limber, 1953), it takes the formCgκ` =∫∆EdE∫ ∞0dzcH(z)1χ(z)2×W g(E,z)W κ(z)Pgδ(k =`χ(z),z),(2.1)where z is the redshift, E is the gamma-ray energy and ∆E the energy bin that is being integrated over,c is the speed of light in the vacuum, H(z) is the Hubble rate, and χ(z) is the comoving distance. Weemploy a flatΛCDM cosmological model with parameters taken from Planck Collaboration XIII (2016).W g and W κ are the window functions that characterize the redshift and energy dependence of thegamma-ray emitters and the efficiency of gravitational lensing, respectively. Pgδ (k,z) is the three-dimensional cross-power spectrum between the gamma-ray emission for a gamma-ray source class andthe matter density δ , with k being the modulus of the wavenumber and ` the angular multipole. Thefunctional form of the window functions and power spectra depend on the populations of gamma-rayemitters and source galaxy distributions under consideration and are described in the following subsec-tions.The quantity measured from the data is the tangential shear cross-correlation function ξ gγt (ϑ). Thiscorrelation function is related to the angular cross-power spectrum by a Hankel transformation:ξ gγt (ϑ) =12pi∫ ∞0d` `J2(`ϑ)Cgκ` , (2.2)24where ϑ is the angular separation in the sky and J2 is the Bessel function of the first kind of order two.2.2.1 Window functionsThe window function describes the distribution of the signal along the line of sight, averaged over alllines of sight.Gravitational lensingFor gravitational lensing the window function is given by (see, e.g., Bartelmann, 2010)W κ(z) =32H20ΩM(1+ z)χ(z)∫ ∞zdz′χ(z′)−χ(z)χ(z′)n(z′) , (2.3)where H0 is the Hubble rate today, ΩM is the current matter abundance in the Universe, and n(z) is theredshift distribution of background galaxies in the lensing data set. The galaxy distribution dependson the data set and redshift selection, as described in Section 2.3.1. The redshift distribution functionn(z) is binned in redshift bins of width ∆z = 0.05. To compute the window function in Eq. (2.3), n(z)is interpolated linearly between those bins. The resulting window functions for KiDS are shown inthe bottom panel of Fig. 2.1. The width of the window function in Fig. 2.1, especially for the 0.1–0.3redshift bin, is due to the leakage of the photometric redshift distribution outside the redshift selectionrange.Gamma-ray emission from dark matterWe consider two processes by which dark matter can create gamma rays: annihilation and decay.The window function for annihilating dark matter reads (Ando et al., 2006; Fornengo et al., 2014)W gann(z,E) =(ΩDMρc)24pi〈σannv〉2mDM2(1+ z)3∆2(z)× dNanndE[E(1+ z)]e−τ[z,E(1+z)] ,(2.4)where ΩDM is the cosmological abundance of dark matter, ρc is today’s critical density of the Universe,mDM is the rest mass of the dark matter particle, and 〈σannv〉 denotes the velocity-averaged annihilationcross-section, assumed here to be the same in all haloes. The optical depth of gamma rays is given by τ .dNann/dE indicates the number of photons produced per annihilation as a function of photon energy, andsets the gamma-ray energy spectrum. We will consider it to be given by the sum of two contributions:prompt gamma-ray production from dark matter annihilation, which provides the bulk of the emissionat low masses; and inverse Compton scattering of highly energetic dark matter-produced electrons andpositrons on CMB photons, which upscatter in the gamma-ray band. The final states of dark matterannihilation are computed by means of the PYTHIA Monte Carlo package v8.160 (Sjo¨strand et al.,2008). The inverse Compton scattering contribution is calculated as in Fornasa et al. (2013), whichassumes negligible magnetic field and no spatial diffusion for the produced electrons and positrons.[Mpc−1cm−2s−1sr−1] ×10−11Decaying DMHIGHMIDLOWAstro0.0 0.5 1.0 1.5 2.0z−κ(z)[Mpc−1]×10−5z: 0.1 → 0.3z: 0.3 → 0.5z: 0.5 → 0.7z: 0.7 → 0.9z: 0.1 → 0.9Figure 2.1: Top: window functions for the gamma-ray emissions W g for the energy range 0.5–500GeV and redshift selection of 0.1–0.9. Shown are the window functions for the three annihi-lating dark matter scenarios considered, i.e., HIGH (blue), MID (purple), LOW (red); decayingdark matter (black); and the sum of the astrophysical sources (green). The annihilation sce-narios assume mDM = 100 GeV and 〈σannv〉= 3×10−26 cm3s−1. For decaying dark matter,mDM = 200 GeV and Γdec = 5× 10−28 s−1. The predictions for annihilating and decayingdark matter are for the bb¯ channel. We consider three populations of astrophysical sourcesthat contribute to the UGRB: blazars, mAGN, and SFG, described in Section 2.2.1. Bottom:the lensing window functions for the five tomographic bins chosen for KiDS.Results will be shown for three final states of the annihilation: bb¯ pairs, which yields a relatively softspectrum of photons and electrons, mostly associated with hadronisation into pions and their subsequentdecay; µ+µ−, which provides a relatively hard spectrum, mostly associated with final state radiation ofphotons and direct decay of the muons into electrons; and τ+τ−, which lies between the first two cases,being a leptonic final state, but with semi-hadronic decay into pions (Fornengo et al., 2004; Cembranoset al., 2011; Cirelli et al., 2011).The optical depth τ in Eq. (2.4) accounts for attenuation of gamma rays due to scattering off theextragalactic background light (EBL), and is taken from Franceschini et al. (2008).The clumping factor ∆2 is related to how dark matter density is clustered in haloes and subhaloes.Its definition depends on the square of the dark matter density; therefore, it is a measure of the amount26of annihilations happening and, thus, the expected gamma-ray flux. The clumping factor is defined as(see, e.g., Fornengo et al., 2014)∆2(z)≡ 〈ρ2DM〉ρ¯2DM=∫ MmaxMmindMdnhdM(M,z) [1+bsub(M,z)]×∫d3xρ2h (~x|M,z)ρ¯2DM,(2.5)where ρ¯DM is the current mean dark matter density, dnh/dM is the halo mass function (Sheth et al.,1999), Mmin is the minimal halo mass, Mmax is the maximal mass of haloes (for definiteness, we use1018 M, but the results are insensitive to the precise value assumed), ρh(x|M,z) is the dark matterdensity profile of a halo with mass M at redshift z, taken to follow a Navarro-Frenk-White (NFW)profile (Navarro et al., 1997), and bsub is the boost factor that encodes the ‘boost’ to the halo emissionprovided by subhaloes. To characterize the halo profile and the subhalo contribution, we need to specifytheir mass concentration. Modelling the concentration parameter c(M,z) at such small masses and forsubhaloes is an ongoing topic of research and is the largest source of uncertainty of the models in thisanalysis. We consider three cases: LOW, which uses the concentration parameter derived in Prada et al.(2012), (see also Sa´nchez-Conde et al., 2014); HIGH, based on Gao et al. (2012); and MID, followingthe recent analysis of Moline´ et al. (2017). The last one represents our reference case, with predictionsthat are normally intermediate between those of the LOW and HIGH. The authors in Moline´ et al. (2017)refined the estimation of the boost factor of Sa´nchez-Conde et al. (2014) by modelling the dependenceof the concentration of the subhaloes on their position in the host halo. Accounting for this dependenceand related effects, such as tidal stripping, leads to an increase of a factor 1.7 in the overall boost factorover the LOW model. Predictions for the dark matter clumping factor for the three models are shown inFig. 2.2.Since the number of subhaloes and, therefore, the boost factor, increases with increasing host halomass, the integral in Eq. (2.5) is dominated group and cluster-sized haloes (Ando et al., 2013). However,in the absence of subhaloes, the clumping factor in Eq. (2.5) strongly depends on the low-mass cutoffMmin, since the mass integral in Eq. (2.5) diverges as Mmin→ 0 (Ando et al., 2006). The minimum halomass Mmin depends on the free-streaming scale of dark matter, which is assumed to be in the range of10−12–10−3 M (Profumo et al., 2006; Bringmann, 2009). We therefore choose an intermediary masscutoff of Mmin = 10−6 M. As all our models include substructure, the dependence on Mmin is at mostO(1). Specifically, the changes to clumping factor due to varying Mmin in the range 10−12–10−2 Mare less than the uncertainty in the concentration parameter (Regis et al., 2015).The window function of decaying dark matter is given by (Ando et al., 2006; Ibarra et al., 2013;Fornengo et al., 2014)W gdec(z,E) =ΩDMρc4piΓdecmDMdNdecdE[E(1+ z)]e−τ[z,E(1+z)] , (2.6)where Γdec is the decay rate and dNdecdE (E) =dNanndE (2E), i.e., the energy spectrum for decaying dark matter270.0 0.5 1.0 1.5 2.0 2.5 3.0z104105106Clumpingfactor∆2(z)HIGHMIDLOWFigure 2.2: Dark matter clumping factor ∆2, as defined in Eq. (2.5), as a function of redshift forthe LOW (dash-dotted red), MID (dashed purple) and HIGH (solid blue) scenarios of darkmatter clustering. The MID model is built from its expression at z= 0 in Moline´ et al. (2017),assuming the same redshift scaling as in Prada et al. (2012).is the same as that for annihilating dark matter described above, at twice the energy (Cirelli et al., 2011).Unlike annihilating dark matter, decaying dark matter does not depend on the clumping factor and theexpected emission is thus much less uncertain. A set of representative window functions for annihilatingand decaying dark matter is shown in the top panel of Fig. 2.1. In Fig. 2.3 we show the average all-skygamma-ray emission expected from annihilating dark matter for the three clumping scenarios describedabove and from decaying dark matter.Gamma-ray emission from astrophysical sourcesBesides dark matter, gamma rays are produced by astrophysical sources, which will contaminate, andeven dominate over the expected dark matter signal. Indeed, astrophysical sources have been shown tobe able to fully explain the observed cross-correlations between gamma rays and tracers of LSS, likegalaxy catalogues (Xia et al., 2015; Cuoco et al., 2015). For this analysis, we model three populationsof astrophysical sources of gamma rays: blazars; misaligned AGN (mAGN); and star forming galaxies(SFGs). Blazars are likely AGN with their jets directly pointed towards us, while mAGN have jetspointed in our general direction, resulting in lower luminosities than blazars. The sum of the gamma-rayemissions produced by the three extragalactic astrophysical populations described above approximatelyaccounts for all the UGRB measured (see Fornasa et al., 2015), as shown in Fig. 2.3, where the emissionsfrom the three astrophysical source classes are compared to the most recent measurement of the UGRBenergy spectrum from Ackermann et al. (2015a). For each of these astrophysical gamma-ray sources,28100 101 102E [GeV]10−1010−910−810−710−6E2dI/dE[GeVcm−2s−1sr−1]Decaying DMHIGHMIDLOWAstroFermi URGBFigure 2.3: Intensities of the gamma-ray source classes considered in this work: annihilating darkmatter assuming HIGH (solid blue), MID (solid purple), LOW (solid red) clustering models;decaying dark matter (solid black); and astrophysical sources (‘Astro’, solid green). Thedark matter particle properties are the same as in Fig. 2.1. The astrophysical sources arefurther divided into blazars (dashed green), mAGN (dotted green), and SFG (dash-dottedgreen). The black data points represents the observed isotropic component of the UGRB(Ackermann et al., 2015a).we consider a window function of the formW gS(z,E) = χ(z)2∫ Lmax(Fsens,z)LmindLdFdE(L ,z)Φ(L ,z) , (2.7)whereL is the gamma-ray luminosity in the energy interval 0.1–100 GeV, and Φ is the gamma-ray lu-minosity function (GLF) corresponding to one of the source classes of astrophysical emitters included inour analysis. The upper bound,Lmax(Fsens,z), is the luminosity above which an object can be resolved,given the detector sensitivity Fsens, taken from Ackermann et al. (2015b). As we are interested in thecontribution from unresolved astrophysical sources, only sources with luminosities smaller than Lmaxare included. Conversely, the minimum luminosity Lmin depends on the properties of the source classunder investigation. The differential photon flux is given by dF/dE = dNS/dE × e−τ[z,E(1+z)], wheredNS/dE is the observed energy spectrum of the specific source class and the exponential factor againaccounts for the attenuation of high-energy photons by the EBL.We consider a unified blazar model combining BL Lacertae and flat-spectrum radio quasars as asingle source class. The GLF and energy spectrum are taken from Ajello et al. (2015) where they arederived from a fit to the properties of resolved blazars in the third Fermi-LAT catalogue (Acero et al.,2015).In the case of mAGN, we follow Di Mauro et al. (2014), who studied the correlation between thegamma-ray and radio luminosity of mAGN, and derived the GLF from the radio luminosity function.29We consider their best-fittingL –Lr,core relation and assume a power-law spectrum with index αmAGN =2.37.To build the GLF of SFG, we start from the IR luminosity function of Gruppioni et al. (2013), addingup spiral, starburst, and SF-AGN populations of their table 8, and adopt the best-fittingL –LIR relationfrom Ackermann et al. (2012). The energy spectrum is taken to be a power law with spectral indexαSFG = 2.7.The window function and average all-sky emission expected from the astrophysical sources areshown as green lines in the top panel of Fig. 2.1 and Fig. 2.3, respectively.2.2.2 Three-dimensional power spectrumThe three-dimensional cross-power spectrum Pgδ between the gamma-ray emission of a source class gand the matter density is defined as〈 fˆg(z,~k) fˆ ∗δ (z′,~k′)〉= (2pi)3δ 3(~k+~k′)Pgδ (k,z,z′) , (2.8)where fˆg and fˆδ denote the Fourier transform of the emission of the specific class of gamma-ray emittersand matter density, respectively, and 〈 .〉 indicates the average over the survey volume. Using the Limberapproximation, one can set z = z′ in Eq. 2.8. The density of gamma-ray emission due to decaying darkmatter traces the dark matter density contrast δDM, while the emission associated with annihilating darkmatter traces δ 2DM. Astrophysical sources are assumed to be point-like biased tracers of the matterdistribution. Finally, lensing directly probes the matter contrast δM. To compute Pgδ , we follow the halomodel formalism (for a review, see, e.g., Cooray et al., 2002), and write P = P1h +P2h. We derive the1-halo term P1h and the two-halo term P2h as in Fornengo et al. (2015) and in Camera et al. (2015).Dark matter gamma-ray sourcesThe 3D cross-power spectrum between dark matter sources of gamma rays and matter density is givenbyP1hgDMκ(k,z) =∫ MmaxMmindMdnhdM(M,z) vˆgDM(k|M,z) uˆκ(k|M,z) ,P2hgDMκ(k,z) =[∫ MmaxMmindMdnhdM(M,z)bh(M,z) vˆgDM(k|M,z)]×[∫ MmaxMmindMdnhdM(M,z)bh(M,z)uˆκ(k|M,z)]×Plinδδ (k,z) ,(2.9)where Plinδδ is the linear matter power spectrum, bh is the linear bias (taken from the model of Sheth et al.,1999), and uˆκ(k|M,z) is the Fourier transform of the matter halo density profile, i.e., ρh(~x|M,z)/ρ¯DM.The Fourier transform of the gamma-ray emission profile for dark matter haloes is described byvˆgDM(k|M,z). For decaying dark matter, vˆgDM = uˆκ , i.e., the emission follows the dark matter density30profile. Conversely, the emission for annihilating dark matter follows the square of the dark matter den-sity profile: vˆgDM(k|M,z) = uˆann(k|M,z)/∆(z)2, where uˆann is the Fourier transform of the square of themain halo density profile plus its substructure contribution, and ∆(z)2 is the clumping factor. The masslimits are Mmin = 10−6 M and Mmax = 1018 M again.Astrophysical gamma-ray sourcesThe cross-correlation of the convergence with astrophysical sources is sourced by the 3D power spec-trumP1hgSκ(k,z) =∫ LmaxLmindLΦ(L ,z)〈 fS〉dFdE(L ,z) uˆκ(k|M(L ,z),z) ,P2hgSκ(k,z) =[∫ LmaxLmindL bS(L ,z)Φi(L ,z)〈 fS〉dFdE(L ,z)]×[∫ MmaxMmindMdnhdMbh(M,z)uˆκ(k|M,z)]×Plinδδ (k,z) ,(2.10)where bS is the bias of gamma-ray astrophysical sources with respect to the matter density, for whichwe adopt bS(L ,z) = bh(M(L ,z)). That is, a source with luminosity L has the same bias bh as a halowith mass M, with the relation M(L ,z) between the mass of the host halo M and the luminosity of thehosted objectL taken from Camera et al. (2015). The mean flux 〈 fS〉 is defined as 〈 fS〉=∫dL dFdEΦ.2.3 Data2.3.1 Weak lensing data setsFor this study we combine CFHTLenS1 and RCSLenS2 data sets from the Canada France Hawaii Tele-scope (CFHT) and KiDS3 from the VLT Survey Telescope (VST), all of which have been optimisedfor weak lensing analyses. The same photometric redshift and shape measurement algorithms havebeen used in the analysis of the three surveys. However, there are slight differences in the algorithmimplementation and in the shear and photometric redshift calibration, as described in the followingsubsections.The sensitivity of the measurement depends inversely on the overlap area between the gamma-ray map and the lensing data, with a weaker dependence on the parameters characterizing the lensingsensitivity, i.e., the galaxy number density and ellipticity dispersion. This is due to the fact that atlarge scales, sampling variance dominates the contribution of lensing to the covariance and reducing theshape noise does not result in an improvement of the overall covariance. This point is further discussedin Section the three surveys, only CFHTLenS and KiDS have full photometric redshift coverage. We chooseto restrict the tomographic analysis to KiDS, as the much smaller area of CFHTLenS is expected toyield a much lower sensitivity for this measurement. In Section 2.5.2 we find that tomography does notappreciably improve the exclusion limits on the dark matter parameters. We thus do not lose sensitivityby restricting the tomographic analysis to KiDS in this work.CFHTLenSCFHTLenS spans a total area of 154 deg2 from a mosaic of 171 individual MEGACAM pointings,divided into four compact regions (Heymans et al., 2012). Details of the data reduction are describedin Erben et al. (2013). The observations in the five bands ugriz of the survey allow for the precisemeasurement of photometric redshifts (Hildebrandt et al., 2012). The shape measurement with LENSFITis described in detail in Miller et al. (2013). We make use of all fields in the data set as we are notaffected by the systematics that lead to field rejections in the cosmic shear analyses (Kilbinger et al.,2013). We correct for the additive shear bias for each galaxy individually, while the multiplicative biasis accounted for on an ensemble basis, as described in Section 2.4.1.Individual galaxies are selected based on the Bayesian photometric redshift zB being in the range[0.2, 1.1]. The resulting redshift distribution of the selected galaxies is obtained by stacking the redshiftprobability distribution function of individual galaxies, weighted by the LENSFIT weight. As a result ofthe stacking of the individual redshift PDFs, the true redshift distribution leaks outside the zB selectionrange. Stacking the redshift PDFs can lead to biased estimates of the true redshift distribution of thesource galaxies (Choi et al., 2016) but in light of the large statistical and modelling uncertainties in thisanalysis these biases can be safely neglected here.RCSLenSThe RCSLenS data consist of 14 disconnected regions whose combined total area reaches 785 deg2.A full survey and lensing analysis description is given in Hildebrandt et al. (2016). RCSLenS uses thesame LENSFIT version as CFHTLenS but with a different size prior, as galaxy shapes are measured fromi-band images in CFHTLenS, whereas RCSLenS uses the r-band. The additive and multiplicative shearbiases are accounted for in the same fashion as in CFHTLenS.Multi-band photometric information is not available for the whole RCSLenS footprint, therefore weuse the redshift distribution estimation technique described in Harnois-De´raps et al. (2016) and Hojjatiet al. (2016). Of the three magnitude cuts considered in Hojjati et al. (2016), we choose to select thesource galaxies such that 18<magr < 26, as this selection yielded the strongest cross-correlation signalin Hojjati et al. (2016). This cut is close to the 18 < magr < 24 in Harnois-De´raps et al. (2016) but withthe faint cutoff determined by the shape measurement algorithm. The redshift distribution is derivedfrom the CFHTLenS-VIPERS sample (Coupon et al., 2015), a UV and IR extension of CFHTLenS.We stack the redshift PDF in the CFHTLenS-VIPERS sample, accounting for the RCSLenS magnitudeselection, r-band completeness, and galaxy shape measurement (LENSFIT) weights.320. 2.4: Photon count map of the gamma-ray sky for the energy range 0.5–500 GeV forultracleanveto photons.KiDSThe third data set considered here comes from KiDS, which currently covers 450 deg2, with completeugri four band photometry in five patches. Galaxy shapes are measured in the r-band using the newself-calibrating LENSFIT (Fenech Conti et al., 2017). Cross-correlation studies such as this work areonly weakly sensitive to additive biases and, being linear in the shear, are less affected by multiplica-tive biases than cosmic shear studies. Nonetheless, the analysis still benefits from well-calibrated shapemeasurements. The residual multiplicative shear bias is accounted for on an ensemble basis, as forCFHTLenS and RCSLenS. To correct for the additive bias we subtract the LENSFIT weighted ellip-ticity means in each tomographic bin. A full description of the survey and data products is given inHildebrandt et al. (2017).We select galaxies with 0.1≤ zB < 0.9 and then further split the data into four redshift bins [0.1, 0.3],[0.3, 0.5], [0.5, 0.7], and [0.7, 0.9]. We derive the effective n(z) following the DIR method introducedin Hildebrandt et al. (2017).2.3.2 Fermi-LATFor this work we use Fermi-LAT data gathered until 2016 September 1, spanning over eight years ofobservations. We use Pass 8 event reconstruction and reduce the data using FERMI SCIENCE TOOLSversion v10r0p5. We select FRONT+BACK converting events (evtype=3) between energies of 0.5and 500 GeV. We restrict our main analysis to ultracleanveto photons (evclass=1024). Weverify that selecting clean photons (evclass=256) does not change the results of the analysis.Furthermore, we apply the cuts (DATA_QUAL>0)&&(LAT_CONFIG==1) on the data quality. Wethen create full sky HEALPIX4 photon count and exposure maps with nside=1024 (Go´rski et al.,4http://healpix.sourceforge.net/332005) in 20 logarithmically spaced energy bins in the range mentioned above. The photon count mapfor the whole energy range is shown in Fig. 2.4.The flux map used in the cross-correlation analysis is obtained by dividing the count maps by theexposure maps in each energy bin before adding them. We have confirmed that the energy spectrum ofthe individual flux maps follows a broken power law with an index of 2.34±0.02, consistent with thatobtained in previous studies of the UGRB (Ackermann et al., 2015d).We also create maps for four energy bins 0.5–0.766 GeV, 0.766–1.393 GeV, 1.393–3.827 GeV, and3.827–500 GeV. The bins are chosen such that they would contain equal photon counts for a power lawspectrum with index 2.5. The flux maps for the four energy bins are computed by first dividing eachenergy bin into three logarithmically spaced bins, creating flux maps for these fine bins, and then addingthem up.The total flux is dominated by resolved point sources and, to a lesser extent, by diffuse Galacticemission. To probe the unresolved component of the gamma-ray sky, we mask the 500 brightest pointsources in the third Fermi point source catalogue Acero et al. (2015) with circular masks having a radiusof two degrees. The remaining point sources are masked with one degree circular masks. We checkedthat the analysis is robust with respect to other masking strategies. The effect of the diffuse Galacticemission is minimized by subtracting the gll_iem_v06 model. Furthermore, we employ a 20◦ cutin Galactic latitude. It has been shown in Shirasaki et al. (2016) that this cross-correlation analysis isrobust against the choice made for the model of diffuse Galactic emission. We have confirmed that ourresults are not significantly affected even in the extreme scenario of not removing the diffuse Galacticemission at all. This represents an important benefit of using cross-correlations to study the UGRB overstudies of the energy spectrum alone, as in Ackermann et al. (2015d).The robustness of these selection and cleaning choices is demonstrated in Fig. 2.6, where the impactof the event selection, point source masks, and cleaning of the diffuse Galactic emission on the cross-correlation signal is shown. None of these choices lead to a significant change in the measured correla-tion signal, highlighting the attractive feature of cross-correlation analyses that uncorrelated quantities,such as Galactic emissions and extragalactic effects like lensing, do not bias the signal.The point spread function (PSF) of Fermi-LAT is energy dependent and, especially at low ener-gies, significantly reduces the cross-correlation signal power at small angular scales, as demonstrated inFig. 2.5. The pixelation of the gamma-ray sky has a similar but much weaker effect. In this analysis, wechoose to account for this suppression of power by forward modelling. That is, rather than correctingthe measurements, the predicted angular power spectra Cgκ` are modified to account for the effect of thePSF and pixel window function. The PSF provided by FERMI SCIENCE TOOLS takes into account allour photon selection criteria and is sufficiently accurate for the sensitivity in this analysis.5The gamma-ray data used in the analysis are obtained by cutting out regions around the lensingfootprints. To increase the sensitivity at large angular scales, we include an additional four degree wideband around each of the 23 lensing patches.5Systematic uncertainty on the PSF: https://fermi.gsfc.nasa.gov/ssc/data/analysis/LAT caveats.html34102 103`10−1310−1210−1110−1010−9`(`+1)/2piCgκ`[cm−2s−1sr−1]Decaying DMHIGHMIDLOWAstroFigure 2.5: Model Cgκ` for three annihilating dark matter scenarios (HIGH, MID, and LOW), decay-ing dark matter, and astrophysical sources (‘Astro’). The models and formatting are the sameas in Figs. 2.1 and 2.3. The models assume the n(z) for the z∈ [0.1,0.9] bin for KiDS and theenergy range 0.5–500 GeV. The effect of the Fermi-LAT PSF on the cross-power spectrumis illustrated by dashed lines for the four energy bins, with the lowest energy bin having thestrongest suppression of power at small scales.2.4 Methods2.4.1 EstimatorsTo measure the cross-correlation function between gamma rays and lensing, we employ the tangentialshear estimator (see also Shirasaki et al., 2014; Harnois-De´raps et al., 2016; Hojjati et al., 2016):ξˆ gγt/x(ϑ) =∑i j wiet/xi j g j∆i j(ϑ)∑i j wi∆i j(ϑ)11+K(ϑ),11+K(ϑ)=∑i j wi∆i j(ϑ)∑i j wi(1+mi)∆i j(ϑ),(2.11)where the sum runs over all galaxies i and pixels j of the gamma-ray flux map, wi is the LENSFIT-weightof galaxy i and et/xi j is the tangential (t) or cross (x) component of the shear with respect to the positionof pixel j, g j is the flux at pixel j, and ∆i j(ϑ) accounts for the angular binning, being equal to 1 if thedistance between galaxy i and pixel j falls within the angular bin centred on ϑ and 0 otherwise. Thefactor of 11+K accounts for the multiplicative shear bias, with mi being the multiplicative shear bias ofgalaxy i. We choose to work with the gamma-ray flux instead of its contrast (g−〈g〉)/〈g〉, since themean flux 〈g〉 would be encumbered by large systematic uncertainties due to to imperfect subtraction oflarge-scale Galactic emission and leakage of flux from masked resolved point sources.The ξˆ gγt/x(ϑ) measurement described in Eq. (2.11) exhibits strong correlation between the angular35200 400 600 800 1000 1200 1400`−1.0−`(`+1)/2piCgγ`×10−9z: 0.1-0.9E: 0.5-500 GeVγEγE big masksγE no DGE cleaningγE event type: cleanFigure 2.6: Measurement of the cross-spectrum Cˆgκ` between Fermi-LAT gamma rays in the en-ergy range 0.5–500 GeV and KiDS weak lensing data in the redshift range 0.1–0.9 for dif-ferent gamma-ray data preparation choices: fiducial, as described in Section 2.3.2 (blackpoints); using two-degree radius circular masks for all point sources (red squares); no clean-ing of the diffuse Galactic emission (DGE) (blue stars); and using the clean event selection(green triangles).bins at all scales. This complicates the estimation of the covariance matrix, as the off-diagonal elementshave to be estimated accurately. On the other hand, the covariance of the angular cross-power spec-trum Cˆgκ` is largely diagonal since the measurement is noise-dominated. We thus choose to work withthe angular power spectrum Cˆgκ` instead of the correlation function ξˆgγt (ϑ). Inverting the relation inEq. (2.2), one can construct an estimator for the angular cross-power spectrum Cˆgκ` based on the mea-surement of ξˆ gγt (ϑ) (Schneider et al., 2002; Szapudi et al., 2001b). Specifically, working in the flat-skyapproximation, one can writeCˆgκ` = 2pi∫ ∞0dϑ ϑJ2(`ϑ)ξˆ gγt (ϑ) . (2.12)This estimator yields an estimate for the cross-power spectrum between the gamma rays and the E-modeof the shear field. Replacing the tangential shear γt in Eq. (2.12) with the cross component of the shearγx results in an estimate of the cross-power spectrum between the gamma rays and the B-mode of theshear field, which is expected to vanish in the absence of lensing systematics. In Appendix A.1, wecheck that this estimator indeed accurately recovers the underlying power spectrum.To estimate the power spectrum using the estimator in Eq. (2.12), we measure the tangential shearbetween 1 and 301 arc minutes, in 300 linearly spaced bins. The resulting power spectrum is then36binned in five linearly spaced bins between ` of 200 and 1500. At smaller scales the Fermi-LAT PSFsuppresses power, especially at low energies. At very large scales of `. 100, the covariance is affectedby residuals from imperfect foreground subtraction, hence we restrict ourselves to scales of ` > CovariancesOur primary method to estimate the covariance relies on a Gaussian analytical prescription. This isjustified because the covariance is dominated by photon and shape noise, both of which can be modelledaccurately. To verify that this analytical prescription is a good estimate of the true covariance, wecompare it to two internal covariance estimators that estimate the covariance from the data. In the firstapproach, we select random patches on the gamma-ray flux map and correlate them with the lensingdata, as described in Section 2.4.2. For the second method we randomise the pixels of the gamma-rayflux map within the patches used in the cross-correlation measurement, described in Section 2.4.2. Bothof these internal estimators ignore the correlation between gamma rays and lensing, i.e., they implicitlyassume that the Cgκ` term in Eq. (2.13) is zero. Since we do not observe a correlation, this approach isjustified.Unlike an analytical covariance, inverting covariances estimated from a finite number of realizationsincurs a bias (Kaufman, 1967; Hartlap et al., 2007; Taylor et al., 2013; Sellentin et al., 2016). The biasis dependent on the number of degrees of freedom in the measurement. Combining measurements ofmultiple energy or redshift bins increases the size of the measurement vector. Specifically, in the case ofno binning in redshift or energy, the data vector here has five elements, when binning in either redshiftor energy, it contains 20 elements, and when binning in both redshift and energy, its length is 80. Fora fixed number of realizations, the bias therefore changes depending on which data are used in theanalysis, diminishing the advantage gained by combining multiple energy or redshift bins and makingcomparisons between different binning strategies harder. For this reason, we choose the analyticalprescription as our primary method to estimate the covariance.The diagonal elements of the three covariance estimates are shown in Fig. 2.7 for the case of KiDS,showing good agreement between all three approaches. The limits derived from the three covarianceestimations agree as well. Choosing the analytical prescription as our primary method is thus justified.Analytical covarianceWe model the covariance C asC[Cgκ` ] =1fsky(2`+1)∆`(Cˆgg` Cˆκκ` +(Cˆgκ`)2), (2.13)where fsky denotes the fraction of the sky that is covered by the effective area of the survey, ∆` is the`-bin width, Cˆgg` is an estimate of the gamma-ray auto-power spectrum, Cˆκκ` is the convergence auto-power spectrum, and Cˆgκ` is the cross-spectrum between gamma rays and the convergence, calculated asdescribed in Section 2.2. The effective area for the cross-correlation is given by the product of the masksof the gamma-ray map and lensing data, which corresponds to 99, 308, and 362 deg2 for CFHTLenS,3710−3110−3010−2910−28z: 0.1-0.3E: 0.5-0.8 GeVAnalyticRandom patchesRandom fluxz: 0.1-0.3E: 0.8-1.4 GeVz: 0.1-0.3E: 1.4-3.8 GeVz: 0.1-0.3E: 3.8-500.0 GeVz: 0.1-0.3E: 0.5-500.0 GeV10−3110−3010−2910−28z: 0.3-0.5E: 0.5-0.8 GeVz: 0.3-0.5E: 0.8-1.4 GeVz: 0.3-0.5E: 1.4-3.8 GeVz: 0.3-0.5E: 3.8-500.0 GeVz: 0.3-0.5E: 0.5-500.0 GeV10−3110−3010−2910−28Var[Cgκ`][cm−4s−2sr−2]z: 0.5-0.7E: 0.5-0.8 GeVz: 0.5-0.7E: 0.8-1.4 GeVz: 0.5-0.7E: 1.4-3.8 GeVz: 0.5-0.7E: 3.8-500.0 GeVz: 0.5-0.7E: 0.5-500.0 GeV10−3110−3010−2910−28z: 0.7-0.9E: 0.5-0.8 GeVz: 0.7-0.9E: 0.8-1.4 GeVz: 0.7-0.9E: 1.4-3.8 GeVz: 0.7-0.9E: 3.8-500.0 GeVz: 0.7-0.9E: 0.5-500.0 GeV200 460 720 980 1240`10−3110−3010−2910−28z: 0.1-0.9E: 0.5-0.8 GeV200 460 720 980 1240`z: 0.1-0.9E: 0.8-1.4 GeV200 460 720 980 1240`z: 0.1-0.9E: 1.4-3.8 GeV200 460 720 980 1240`z: 0.1-0.9E: 3.8-500.0 GeV200 460 720 980 1240 1500`z: 0.1-0.9E: 0.5-500.0 GeVFigure 2.7: Diagonal elements of the analytical covariance (solid blue), covariance from randompatches (dashed red), and covariance from randomized flux (dot-dashed green) for the fiveenergy and redshift bins for KiDS. All three estimates agree at small scales, while the covari-ance derived from random patches shows a slight excess of variance at large scales.RCSLenS, and KiDS, respectively.The gamma-ray auto-power spectrum Cˆgg` is estimated from the same gamma-ray flux maps as usedin the cross-correlation. We measure the auto-spectra of the five energy bins and the cross-spectrabetween the energy bins using POLSPICE6 in 15 logarithmically spaced `-bins between ` of 30 and 2000.Because the measurement is very noisy at large scales, we fit the measured spectra with a spectrum ofthe formCˆgg` =CP+ c `α , (2.14)where CP is the Poisson noise term, and c and α describe a power-law contribution to account for apossible increase of power at very large scales. The values of the intercept c is consistent with zero inall cases, while best-fitting Poisson noise terms are consistent with a direct estimate based on the mean6http://www2.iap.fr/users/hivon/software/PolSpice/38Table 2.1: Total number of galaxies with shape measurements ngal, effective galaxy number den-sity neff, and ellipticity dispersion σe for CFHTLenS, RCSLenS, and KiDS for the cuts em-ployed in this analysis. We follow the prescription in Heymans et al. (2012) to calculate neffand σe.ngal neff [arcmin−2] σeCFHTLenS 4760606 9.44 0.279RCSLenS 14490842 5.84 0.277KiDS 0.1≤ zB < 0.3 3769174 2.23 0.290KiDS 0.3≤ zB < 0.5 3249574 2.03 0.282KiDS 0.5≤ zB < 0.7 2941861 1.81 0.273KiDS 0.7≤ zB < 0.9 2640577 1.49 0.276KiDS 0.1≤ zB < 0.9 12601186 7.54 0.281number of photon counts, i.e.,CP =〈ng/ε2〉Ωpix, (2.15)where ng is the number of observed photons per pixel, ε the exposure per pixel, and Ωpix the solidangle covered by each pixel (Fornasa et al., 2016). Except for the lowest energies, the observed intrinsicangular auto-power spectrum is sub-dominant to the photon shot noise (Fornasa et al., 2016).The lensing auto-power spectrum is given byCˆκκ` =Cκκ` +σ2eneff, (2.16)where Cκκ` is the cosmic shear signal andσ2eneffis the shape noise term, with σ2e being the dispersion perellipticity component and neff the galaxy number density. These parameters are listed in Table 2.1. Thecosmic shear term Cκκ` is calculated using the halo-model. The two terms in Eq. (2.16) are of similarmagnitude, with the shape noise dominating at small scales and sampling variance dominating at largescales. Decreasing σe or increasing neff thus only improves the covariance at scales where the shapenoise makes a significant contribution to Eq. (2.16). However, increasing the area of the lensing surveyand thus the overlap with the gamma-ray map directly decreases the covariance inversely proportionalto the overlap area. For this reason CFHTLenS has a low sensitivity in this analysis, even though it isthe deepest survey of the three. Although RCSLenS has the largest effective area, the covariance forKiDS is slightly smaller, since the increase in depth is large enough to overcome the area advantage ofRCSLenS.Random patchesWe select 100 random patches from the gamma-ray map as an approximation of independent realizationsof the gamma-ray sky. The patches match the shape of the original gamma-ray cutouts, i.e., the lensing39footprints plus a four degree wide band, but have their positions and orientations randomised. Thepatches are chosen such that they do not lie within the Galactic latitude cut.These random patches are uncorrelated with the lensing data but preserve the auto-correlation of thegamma rays and hence account for sampling variance in the gamma-ray sky, including residuals of theforeground subtraction.For small patches, the assumption of independence is quite accurate, as the probability of two ran-dom patches overlapping is low. Larger patches will correlate to a certain degree. This lack of inde-pendence might lead to an underestimation of the covariance. This correlation is minimized by rotatingeach random patch, making the probability of having two very similar patches low.The diagonal elements of the resulting covariance are shown in Fig. 2.7. While the agreement withthe Gaussian covariance is good at small scales, there is an excess of variance at large scales for someenergy and redshift bins. This excess can be explained by a large-scale modulation of the power in thegamma-ray map, which would be sampled by the random patches. This interpretation is consistent withthe strong growth of the error bars of the gamma-ray auto-correlation towards large scales. However,the results of the analysis are not affected significantly by this.Randomized fluxIn a further test of the analytical covariance in Eq. (2.13), we randomize the gamma-ray pixel positionswithin each patch. This preserves the one-point statistics of the flux while destroying any spatial correla-tions. This approach is similar to the random Poisson realizations used in Shirasaki et al. (2014), but weuse the actual one-point distribution of the data themselves instead of assuming a Poisson distribution.Because the pixel values are not correlated anymore, contributions to the large-scale variance dueto imperfect foreground subtraction or leakage of flux from point sources outside of their masks areremoved.The covariance derived from 100 such random flux maps is in good agreement with both the analyt-ical covariance and the covariance estimated from random patches, as shown in Fig. Statistical methodsThe likelihood function we employ to find exclusion limits on the annihilation cross-section 〈σannv〉 ordecay rate Γdec and WIMP mass mDM is given byL (~α|~d) ∝ e− 12 χ2(~d,~α) , (2.17)withχ2(~d,~α) =(~d−~µ(~α))TC−1(~d−~µ(~α)), (2.18)where ~d denotes the data vector, ~µ(~α) the model vector, ~α the parameters considered in the fit, i.e.,either the cross-section 〈σannv〉 and the particle mass mDM or the decay rate Γdec and mDM. The ampli-40200 460 720 980 1240`−1.5−1.0−`(`+1)/2piCgκ`[cm−2s−1sr−1]×10−9E: 0.5-0.8 GeVκE κB200 460 720 980 1240`E: 0.8-1.4 GeV200 460 720 980 1240`E: 1.4-3.8 GeV200 460 720 980 1240`E: 3.8-500.0 GeV200 460 720 980 1240 1500`E: 0.5-500.0 GeVFigure 2.8: Measurement of the cross-spectrum Cˆgκ` between Fermi-LAT gamma rays and weaklensing data from CFHTLenS for five energy bins for gamma-ray photons (black points). Thecross-spectrum of the gamma rays and CFHTLenS B-modes are depicted as red data points.tude of the cross-correlation signal expected from astrophysical sources is kept fixed and thus does notcontribute as an extra free parameter. Finally, C−1 is the inverse of the data covariance.The limits on 〈σannv〉 and Γdec correspond to contours of the likelihood surface described byEq. (2.17). Specifically, for a given confidence interval p, the contours are given by the set of parameters~αcont. for whichχ2(~d,~αcont.) = χ2(~d,~αML)+∆χ2(p) , (2.19)where ~αML is the maximum likelihood estimate of the parameters 〈σannv〉 or Γdec and mDM, χ2 is givenby Eq. (2.18), and ∆χ2(p) corresponds to the quantile function of the χ2-distribution. For this analysiswe are dealing with two degrees of freedom and require 2σ contours, hence ∆χ2(0.95) = 6.18.This approach to estimate the exclusion limits follows recent studies, such as Shirasaki et al. (2016).It should be noted that deriving the limits on 〈σannv〉 or Γdec for a fixed mass mDM is also common inthe literature, see e.g., Fornasa et al. (2016) for a recent example. This corresponds to calculating thequantile function ∆χ2(p) for only one degree of freedom.Care has to be taken when using data-based covariances, such as the random patches and randomizedflux, as the inverse of these covariances is biased (Kaufman, 1967; Hartlap et al., 2007). To account forthe effect of a finite number of realizations, the Gaussian likelihood in Eq. (2.17) should be replaced bya modified t-distribution (Sellentin et al., 2016). Alternatively, the effect of this bias on the uncertaintiesof inferred parameter can be approximately corrected (Hartlap et al., 2007; Taylor et al., 2014). Inlight of the large systematic uncertainties in this analysis we opt for the latter approach when using thedata-based covariances.2.5 Results2.5.1 Cross-correlation measurementsWe present the measurement of the cross-correlation of Fermi-LAT gamma rays with CFHTLenS, RC-SLenS, and KiDS weak lensing data in Figs. 2.8, 2.9, and 2.10, respectively. The measurements for41200 460 720 980 1240`−1.0−`(`+1)/2piCgκ`[cm−2s−1sr−1]×10−9E: 0.5-0.8 GeVκE κB200 460 720 980 1240`E: 0.8-1.4 GeV200 460 720 980 1240`E: 1.4-3.8 GeV200 460 720 980 1240`E: 3.8-500.0 GeV200 460 720 980 1240 1500`E: 0.5-500.0 GeVFigure 2.9: Measurement of the cross-spectrum Cˆgκ` between Fermi-LAT gamma rays and weaklensing data from RCSLenS for five energy bins for gamma-ray photons (black points). Thecross-spectrum of the gamma rays and RCSLenS B-modes are depicted as red data points.−1012×10−9z: 0.1-0.3E: 0.5-0.8 GeVκE κBz: 0.1-0.3E: 0.8-1.4 GeVz: 0.1-0.3E: 1.4-3.8 GeVz: 0.1-0.3E: 3.8-500.0 GeVz: 0.1-0.3E: 0.5-500.0 GeV−1012×10−9z: 0.3-0.5E: 0.5-0.8 GeVz: 0.3-0.5E: 0.8-1.4 GeVz: 0.3-0.5E: 1.4-3.8 GeVz: 0.3-0.5E: 3.8-500.0 GeVz: 0.3-0.5E: 0.5-500.0 GeV−1012`(`+1)/2piCgκ`[cm−2s−1sr−1]×10−9z: 0.5-0.7E: 0.5-0.8 GeVz: 0.5-0.7E: 0.8-1.4 GeVz: 0.5-0.7E: 1.4-3.8 GeVz: 0.5-0.7E: 3.8-500.0 GeVz: 0.5-0.7E: 0.5-500.0 GeV−1012×10−9z: 0.7-0.9E: 0.5-0.8 GeVz: 0.7-0.9E: 0.8-1.4 GeVz: 0.7-0.9E: 1.4-3.8 GeVz: 0.7-0.9E: 3.8-500.0 GeVz: 0.7-0.9E: 0.5-500.0 GeV200 460 720 980 1240`−1012×10−9z: 0.1-0.9E: 0.5-0.8 GeV200 460 720 980 1240`z: 0.1-0.9E: 0.8-1.4 GeV200 460 720 980 1240`z: 0.1-0.9E: 1.4-3.8 GeV200 460 720 980 1240`z: 0.1-0.9E: 3.8-500.0 GeV200 460 720 980 1240 1500`z: 0.1-0.9E: 0.5-500.0 GeVFigure 2.10: Measurement of the cross-spectrum Cˆgκ` between Fermi-LAT gamma rays and weaklensing data from KiDS for five energy bins for gamma-ray photons and five redshift bins forKiDS galaxies (black points). The cross-spectrum of the gamma rays and KiDS B-modesare depicted as red data points.42Table 2.2: χ20 values with respect to the hypothesis of a null signal for the measurements of Cˆgκ`shown in Figs. 2.8, 2.9, and 2.10. The number of degrees of freedom is the number of multi-pole bins, i.e., ν = 5 for all measurements.χ20(Cˆgκ` ,ν = 5)Energy bin [GeV] 0.5–0.8 0.8–1.4 1.4–3.8 3.8–500.0 0.5–500.0CFHTLenS 4.49 7.77 3.78 8.43 2.43RCSLenS 6.06 6.75 2.39 6.47 3.19KiDS 0.1≤ zB < 0.3 5.96 1.85 6.53 6.89 8.47KiDS 0.3≤ zB < 0.5 1.84 1.94 2.75 3.42 2.77KiDS 0.5≤ zB < 0.7 3.27 1.89 4.02 2.56 5.57KiDS 0.7≤ zB < 0.9 4.82 11.42 4.98 2.88 8.76KiDS 0.1≤ zB < 0.9 7.16 1.81 5.42 3.05 6.55CFHTLenS and RCSLenS use a single redshift bin and the five energy bins described in Section 2.3.2.The measurements for KiDS use the same energy bins but are further divided into the five redshift binsgiven in Section 2.3.1.Beside the cross-correlation of the gamma rays and shear due to gravitational lensing (denoted byblack circles), we also show the cross-correlation between gamma rays and the B-mode of the shearas red squares. The B-mode of the shear is obtained by rotating the galaxy orientations by 45◦, whichdestroys the gravitational lensing signal. Any significant B-mode signal would be indicative of spurioussystematics in the lensing data.The χ20 values of the measurements with respect to the hypothesis of a null signal, i.e., ~µ = 0, arelisted in Table 2.2. The χ20 values are consistent with a non-detection of a cross-correlation for allmeasurements. This finding is in agreement with the previous studies Shirasaki et al. (2014, 2016) ofcross-correlations between gamma rays and galaxy lensing. For a 3 σ detection of the cross-correlationwith astrophysical sources,7 the error bars would have to shrink by a factor of 3 with respect to thecurrent error bars for KiDS. This corresponds to a ∼4000 deg2 survey with KiDS characteristics, com-parable in size to the galaxy surveys used in Xia et al. (2015). This is further illustrated in Fig. 2.11,which shows the measurement for KiDS for the unbinned case in comparison with the expected cor-relation signal from astrophysical sources and annihilating dark matter for the HIGH scenario and〈σannv〉 = 3× 10−26 cm3s−1 for mDM = 100 GeV and the bb¯ channel. While these signals are notobservable at current sensitivities, they are within reach of upcoming surveys, such as DES.8The B-mode signal is consistent with zero for all measurements. This suggests that the systematicsthat could introduce B-modes into the measurement are under control. At very small scales, lens-sourceclustering can cause a suppression of the lensing signal (van Uitert et al., 2011; Hoekstra et al., 2015).The angular scales we are probing in this analysis are however not affected by this.7Cross-correlations between tracers of LSS and gamma rays have already been detected in Xia et al. (2015). A significantsignal in the case of future weak lensing surveys is therefore a reasonable expectation.8https://www.darkenergysurvey.org/43200 400 600 800 1000 1200 1400`−0.4−`(`+1)/2piCgκ`[cm−2s−1sr−1]×10−9HIGHAstroKiDSFigure 2.11: Measurement of the cross-spectrum Cˆgκ` between Fermi-LAT gamma rays in the en-ergy range 0.5–500 GeV and KiDS weak lensing data in the redshift range 0.1–0.9 (blackdata points), compared to the expected signal from the sum of astrophysical sources (‘Astro’,solid green) and annihilating dark matter for the HIGH scenario (solid blue). The astrophys-ical sources considered are blazars, mAGN, and SFGs. The annihilating dark matter modelassumes the HIGH scenario, mDM = 100 GeV, and 〈σannv〉= 3×10−26 cm3s−1. The dashedlines show the same models but without correcting for the Fermi-LAT PSF.2.5.2 InterpretationWe wish to exploit the measurements presented in the previous subsection to derive constraints onWIMP dark matter annihilation or decay. To derive the exclusion limits on the annihilation cross-section〈σannv〉 and WIMP mass mDM, and the decay rate Γdec and mDM, we apply the formalism described inSection 2.4.3.In Camera et al. (2015) it was shown that the spectral and tomographic information contained withinthe gamma-ray and lensing data can improve the limits on 〈σannv〉 and Γdec. We show the effect ofdifferent combinations of spectral and tomographic binning for the case of KiDS and annihilations intobb¯ pairs under the HIGH scenario in Fig. 2.12 and for dark matter decay in Fig. 2.13. For these limits weadopt the conservative assumption that all gamma rays are sourced by dark matter, i.e., no astrophysicalcontributions are included. There is a significant improvement of the limits when using four energy binsover a single energy bin, especially at high particle masses mDM. This is due to the fact that the UGRBscales roughly as E−2.3 (Ackermann et al., 2015a). The vast majority of the photons in the 0.5–500 GeVbin therefore come from low energies. However, the peak in the prompt gamma-ray emission inducedby dark matter occurs at energy around mDM/20 (annihilating) or mDM/40 (decaying) for bb¯ and athigher energies for the other channels. Thus, for high mDM, a single energy bin of 0.5–500 GeV largelyincreases the noise without significantly increasing the expected dark matter signal with respect to the3.8–500 GeV bin.The improvement due to tomographic binning is only marginal. Two factors contribute to this lack of44101 102 103mDM [GeV]10−2610−2510−2410−2310−2210−21〈σannv〉[cm3s−1]bb¯, KiDS4z x 4E1z x 4E4z x 1E1z x 1EFigure 2.12: Exclusion limits on the annihilation cross-section 〈σannv〉 and WIMP mass mDM forthe clustering scenarios HIGH (blue), MID (purple), and LOW (red) and for different binningstrategies for the KiDS data. The lines correspond to 2σ upper limits on 〈σannv〉 and mDM,assuming a 100% branching ratio into bb¯. No binning in redshift or energy (1z×1E) isdenoted by dash-dotted lines. The case of binning in redshift but not energy (4z×1E) isplotted as dotted lines, while binning in energy but not redshift (1z×4E) is plotted as dashedlines. Finally, binning in both redshift and energy (4z×4E) is shown as solid lines. Thethermal relic cross-section, from Steigman et al. (2012), is shown in grey.improvement. Firstly, in the case of no observed correlation signal – as is the case here – the differencesin the redshift dependence of the astrophysical and dark matter sources do not come to bear becausethere is no signal to disentangle. Secondly, the lensing window functions are quite broad and thusinsensitive to the featureless window function of the dark matter gamma-ray emissions, as depicted inFig. 2.1. This is due to the cumulative nature of lensing on the one hand and the fact that photo-zcause the true n(z) to be broader than the redshift cuts we impose on the other hand. This is in contrastwith spectral binning, which allows us to sharply probe the characteristic gamma-ray spectrum inducedby dark matter. As shown in Fig. 2.3, annihilating dark matter shows a pronounced pion bump whenannihilating into bb¯ and a cutoff corresponding to the dark matter mass mDM, while for decaying darkmatter the cutoff appears at half the dark matter mass. For this reason we refrain from a tomographicanalysis for CFHTLenS and RCSLenS, as we expect little to no improvements of the limits.The limits can be further tightened by taking into account known astrophysical sources of gammarays. This comes, however, at the expense of introducing new uncertainties in the modelling of saidastrophysical sources. Going forward, we include the astrophysical sources to show the sensitivity reachof such analyses but also show the conservative limits derived under the assumption that all gamma raysare sourced by dark matter.To account for the astrophysical sources, we subtract the combination of the three populations(blazars, mAGN, and SFG) described in Section 2.2 from the observed cross-correlation signal. Thedark matter limits are then obtained by proceeding as before but using the residuals between the cross-45101 102 103mDM [GeV]10−2710−2610−2510−24Γdec[s−1]bb¯, KiDS4z x 4E1z x 4E4z x 1E1z x 1EFigure 2.13: Exclusion limits on the decay rate Γdec and WIMP mass mDM for the bb¯ channelfor different binning strategies for the KiDS data. The style of the lines is analogous toFig. 2.12.correlation measurement and the astrophysical contribution. Since we assume no error on the astro-physical models, the limits obtained by including blazars, mAGN, and SFG contributions should beconsidered as a sensitivity forecast for a future situation where gamma-ray emission from these astro-physical sources will be perfectly understood.The resulting 2σ exclusion limits on the dark matter annihilation cross-section 〈σannv〉 for thebb¯, µ−µ+, and τ−τ+ channels are shown in Fig. 2.14. Finally, the combined exclusion limits forCFHTLenS, RCSLenS, and KiDS are shown in Fig. 2.15 and Fig. 2.16 for annihilating and decayingWIMP dark matter, respectively. The exclusion limits for annihilating dark matter should be comparedto the thermal relic cross-section (Steigman et al., 2012), shown in grey. Under optimistic assumptionsabout the clustering of dark matter, i.e., the HIGH model, and accounting for contributions from astro-physical sources (dashed blue line), we can exclude the thermal relic cross-section for masses mDM . 20GeV for the bb¯ channel. In the case of annihilations or decays into muons or tau leptons, the exclusionlimits change shape and become stronger for large dark matter masses, compared to the b channel. Thisis due to the fact that, for heavy dark matter candidates, inverse Compton scattering produces a signifi-cant amount of gamma-ray emission in the upper energy range probed by our measurement (Ando et al.,2016). If we make the conservative assumption that only dark matter contributes to the UGRB, i.e., wedo not account for the astrophysical sources of gamma rays, the exclusion limits weaken slightly, asseen in the difference between the dashed and solid blue lines in Fig. 2.15. In this case the thermal reliccross-section can be excluded for mDM . 10 GeV for the bb¯ channel. These limits are consistent withthose forecasted in Camera et al. (2015).The exclusion limits when dark matter is assumed to be the only contributor to the UGRB arecomparable to those derived from the energy spectrum of the UGRB in Ackermann et al. (2015d).However, when the contribution from astrophysical sources is accounted for, the limits in Ackermann4610−2610−2510−2410−2310−2210−21〈σannv〉[cm3s−1]bb¯, CFHTLenSHIGHHIGH + astroMIDLOWµ−µ+, CFHTLenS τ−τ+, CFHTLenS10−2610−2510−2410−2310−2210−21〈σannv〉[cm3s−1]bb¯, RCSLenS µ−µ+, RCSLenS τ−τ+, RCSLenS101 102 103mDM [GeV]10−2610−2510−2410−2310−2210−21〈σannv〉[cm3s−1]bb¯, KiDS102 103mDM [GeV]µ−µ+, KiDS102 103mDM [GeV]τ−τ+, KiDSFigure 2.14: Exclusion limits on the annihilation cross-section 〈σannv〉 and WIMP mass mDM at2σ significance for CFHTLenS, RCSLenS, and KiDS and annihilation channels bb¯, µ−µ+,and τ−τ+. CFHTLenS and RCSLenS use four energy bins while KiDS additionally makesuse of four redshift bins. The exclusion limits are for the three clustering scenarios HIGH(blue), MID (purple), and LOW (red). The dashed blue line indicates the improvement of thelimits for the HIGH scenario when including the astrophysical sources in the analysis.et al. (2015d) improve by approximately one order of magnitude, while our limits see only modestimprovements. This is due to the fact that we do not observe a cross-correlation signal. The constrainingpower therefore largely depends on the size of the error bars. The contribution from astrophysicalsources is small compared to the size of our error bars, as shown in Fig. 2.11, explaining the modestgain in constraining power when including the astrophysical sources compared to probes that observea signal. The exclusion limits obtained in Fornasa et al. (2016) from the measurement of the UGRBangular auto-power spectrum are stronger than the ones presented here. Those limits are dominatedby the emission from dark matter subhaloes in the Milky Way, a component that is not considered inour analysis since it does not correlate with weak lensing. When restricting the analysis of the auto-spectrum in Fornasa et al. (2016) to only the extragalactic components, our cross-correlation analysisyields more stringent limits. The limits presented here are comparable to those of similar analyses ofthe cross-correlation between gamma rays and weak lensing (Shirasaki et al., 2014, 2016) but weakerthan those derived from cross-correlations between gamma rays and galaxy surveys (Cuoco et al., 2015;47101 102 103mDM [GeV]10−2610−2510−2410−2310−2210−21〈σannv〉[cm3s−1]bb¯, CFHTLenS + RCSLenS + KiDSHIGHHIGH + astroMIDLOW102 103mDM [GeV]µ−µ+, CFHTLenS + RCSLenS + KiDS102 103mDM [GeV]τ−τ+, CFHTLenS + RCSLenS + KiDSFigure 2.15: Exclusion limits on the annihilation cross-section 〈σannv〉 and WIMP mass mDM at2σ significance for the combination of CFHTLenS, RCSLenS, and KiDS. The style of thelines is the same as for Fig. 2.14.101 102 103mDM [GeV]10−2710−2610−2510−24Γdec[s−1]bb¯, CFHTLenS + RCSLenS + KiDSDecaying DMDecaying DM + astro102 103mDM [GeV]µ−µ+, CFHTLenS + RCSLenS + KiDS102 103mDM [GeV]τ−τ+, CFHTLenS + RCSLenS + KiDSFigure 2.16: Exclusion limits on the decay rate Γdec and WIMP mass mDM at 2σ significance forthe combination of CFHTLenS, RCSLenS, and KiDS (solid black). Including the astro-physical sources in the analysis results in the more stringent exclusion limits denoted by theblack dashed line.Regis et al., 2015). The exclusion limits from all these extragalactic probes are somewhat weaker thanthose derived from dSphs (Ackermann et al., 2015c; Baring et al., 2016).The weaker limits obtained when using KiDS data, compared to those obtained from RCSLenSdata, can be traced to the high data point at small scales in the low energy bins. Restricting the analysisto the `-range of 200 to 1240, i.e., removing the last data point, improves the limits derived from theKiDS to exceed those derived from RCSLenS, as one would expect from the covariances of the twomeasurements. To check whether the high data point is part of a trend that might become significant ateven smaller scales, we extend the measurement to higher ` modes. Doing so reveals a high scatter ofthe data points around zero beyond `& 1500, and no further excess of power at smaller scales. It shouldbe noted that at these small scales, we are probing close to the pixel scale and are within the Fermi-LATPSF, so the signal is expected to be consistent with zero there. Including astrophysical sources absorbssome of the effect of the high data point at small scales. The limits including astrophysical sources ofgamma rays are thus closer to those obtained from RCSLenS than those assuming only dark matter asthe source of gamma rays.482.6 ConclusionWe have measured the angular cross-power spectrum of Fermi-LAT gamma rays and weak gravitationallensing data from CFHTLenS, RCSLenS, and KiDS. Combined together, the three surveys span a totalarea of more than 1000 deg2. We made use of eight years of Pass 8 Fermi-LAT data in the energyrange 0.5–500 GeV which was divided further into four energy bins. For CFHTLenS and RCSLenS, themeasurement was done for a single redshift bin, while the KiDS data were further split into five redshiftbins, making this the first measurement of tomographic weak lensing cross-correlation. We find noevidence of a cross-correlation signal in the multipole range 200 ≤ ` < 1500, consistent with previousstudies and forecasts based on the expected signal and current error bars.Using these measurements we constrain the WIMP dark matter annihilation cross-section 〈σannv〉and decay rate Γdec for WIMP masses between 10 GeV and 1 TeV. Assuming the HIGH model forsmall-scale clustering of dark matter and accounting for astrophysical sources, we are able to excludethe thermal annihilation cross-section for WIMPs of masses up to 20 GeV for the bb¯ channel. Notaccounting for the astrophysical contribution weakens the limits only slightly, while the exclusion limitsfor the more conservative clustering models MID and LOW are a factor of about 10 weaker. We find thattomography does not significantly improve the constraints. However, exploiting the spectral informationof the gamma rays strengthens the limits by up to a factor 3 at high masses.The exclusion limits derived in this work are competitive with others derived from the UGBR, suchas its intensity energy spectrum (Ackermann et al., 2015d), auto-power spectrum (Fornasa et al., 2016),cross-correlation with weak lensing (Shirasaki et al., 2014, 2016) or galaxy surveys (Regis et al., 2015;Cuoco et al., 2015). Exclusion limits derived from local probes, such as dSphs, are stronger, however(Ackermann et al., 2015c).Future avenues to build upon this analysis include the use of upcoming large area lensing data sets,such as future KiDS data, Dark Energy Survey (DES), Hyper Suprime-Cam (HSC)9, Large SynopticSurvey Telescope (LSST)10, and Euclid11, which will make it possible to detect a cross-correlation sig-nal between gamma rays and gravitational lensing. The analysis would also benefit from extending therange of the gamma-ray energies covered, by making use of measurements from atmospheric Cherenkovtelescopes, which are more sensitive to high-energy photons (Ripken et al., 2014).Instead of treating the astrophysical contributions as a contamination to a dark matter signal, themeasurements presented in this work could be used to investigate the astrophysical extragalactic gamma-ray populations that are thought to be responsible for the UGRB. We defer this to a future analysis.9http://www.naoj.org/Projects/HSC/10http://www.lsst.org/11http://sci.esa.int/euclid/49Chapter 3Cross-correlating Planck tSZ withRCSLenS weak lensing: implications forcosmology and AGN feedbackWe present measurements of the spatial mapping between (hot) baryons and the total matter in the Uni-verse, via the cross-correlation between the thermal Sunyaev-Zeldovich (tSZ) map from Planck andthe weak gravitational lensing maps from the Red Sequence Cluster Survey (RCSLenS). The cross-correlations are performed at the map level where all the sources (including diffuse intergalactic gas)contribute to the signal. We consider two configuration-space correlation function estimators, ξ y−κ andξ y−γt , and a Fourier space estimator, Cy−κ` , in our analysis. We detect a significant correlation out tothree degrees of angular separation on the sky. Based on statistical noise only, we can report 13σ and17σ detections of the cross-correlation using the configuration-space y–κ and y–γt estimators, respec-tively. Including a heuristic estimate of the sampling variance yields a detection significance of 7σ and8σ , respectively. A similar level of detection is obtained from the Fourier-space estimator, Cy−κ` . Aseach estimator probes different dynamical ranges, their combination improves the significance of thedetection. We compare our measurements with predictions from the cosmo-OWLS suite of cosmologi-cal hydrodynamical simulations, where different galaxy feedback models are implemented. We find thata model with considerable AGN feedback that removes large quantities of hot gas from galaxy groupsprovides the best match to the measurements.3.1 IntroductionWeak gravitational lensing has matured into a precision tool. The fact that it is insensitive to galaxybias has made lensing a powerful probe of large-scale structure. However, our lack of a completeunderstanding of small-scale astrophysical processes has been identified as a major source of uncertaintyfor the interpretation of the lensing signal. For example, baryonic physics has a significant impact onthe matter power spectrum at intermediate and small scales with k& 1hMpc−1 (van Daalen et al., 2011)and ignoring such effects can lead to significant biases in our cosmological inference (Semboloni et al.,502011; Harnois-De´raps et al., 2015b). On the other hand, if modelled accurately, these effects can beused as a powerful way to probe the role of baryons in structure formation without affecting the abilityof lensing to probe cosmological parameters and the dark matter distribution.One can gain insights into the effects of baryons on the total mass distribution by studying thecross-correlation of weak lensing with baryonic probes. In this way, one can acquire information thatis otherwise inaccessible, or very difficult to obtain, from the lensing or baryon probes individually.Cross-correlation measurements also have the advantage that they are immune to residual systematicsthat do not correlate with the respective signals. This enables the clean extraction of information fromdifferent probes.Recent detections of the cross-correlation between the tSZ signal and gravitational lensing has al-ready revealed interesting insights about the evolution of the density and temperature of baryons aroundgalaxies and clusters. van Waerbeke et al. (2014) found a 6 σ detection of the cross-correlation be-tween the galaxy lensing convergence, κ , from the Canada-France-Hawaii Telescope Lensing Survey(CFHTLenS) and the tSZ signal (y) from Planck. Further theoretical investigations using the halo model(Ma et al., 2015) and hydrodynamical simulations (Hojjati et al. 2015; Battaglia et al. 2015) demon-strated that about 20% of the cross-correlation signal arises from low-mass haloes Mhalo ≤ 1014M, andthat about a third of the signal originates from the diffuse gas beyond the virial radius of haloes. Whilethe majority of the signal comes from a small fraction of baryons within haloes, about half of all baryonsreside outside haloes and are too cool (T ∼ 105 K) to contribute to the measured signal significantly. Wealso note that Hill et al. (2014) presented a correlation between weak lensing of the CMB (as opposedto background galaxies) and the tSZ with a similar significance of detection, whose signal is dominatedby higher-redshift (z > 2) sources than the galaxy lensing-tSZ signal.The galaxy lensing-tSZ cross-correlation studies described above were limited. In van Waerbekeet al. (2014), for example, statistical uncertainty dominates due to the relatively small area of theCFHTLenS survey (∼150 deg2). The tSZ maps were constructed from the first release of the Planckdata. And finally, the theoretical modelling of the cross-correlation signal was not as reliable for com-parison with data as it is today.In this chapter, we use the Red Cluster Survey Lensing (RCSLenS) data (Hildebrandt et al., 2016)and the recently released tSZ maps by the Planck team (Planck Collaboration XXII, 2016). RCSLenScovers an effective area of approximately 560 deg2, which is roughly four times the area covered byCFHTLenS (although the RCSLenS data is somewhat shallower). Combined with the high-quality tSZmaps from Planck, we demonstrate a significant improvement in our measurement uncertainties com-pared to the previous measurements in van Waerbeke et al. (2014). In this chapter, we also utilize anestimator of lensing mass-tSZ correlations where the tangential shear is used in place of the conver-gence. As discussed in Section 3.2.1, this estimator avoids introducing potential systematic errors to themeasurements during the mass map making process and we also show that it leads to an improvementin the detection significance.We compare our measurements to the predictions from the cosmo-OWLS suite of cosmologicalhydrodynamical simulations for a wide range of baryon feedback models. We show that models with51considerable AGN feedback reproduce our measurements best when a WMAP-7yr cosmology is em-ployed. Interestingly, we find that all of the models over-predict the observed signal when a Planckcosmology is adopted. In addition, we also compare our measurements to predictions from the halomodel with the baryonic gas pressure modelled using the so-called ‘universal pressure profile’ (UPP).We find consistency in the cosmological conclusions drawn from the halo model approach with thatdeduced from comparisons to the hydrodynamical simulations.The organization of the chapter is as follows. We present the theoretical background and the data inSection 3.2. The measurements are presented in Section 3.3 and the covariance matrix reconstruction isdescribed in Section 3.4. The implication of our measurements for cosmology and baryonic physics aredescribed in Section 3.5 and we summarize in Section Observational data and theoretical models3.2.1 Cross-correlationFormalismWe work with two lensing quantities in this chapter, the gravitational lensing convergence, κ , and thetangential shear, γt . The convergence, κ(~θ) is given byκ(~θ) =∫ wH0dwW κ(w)δm(~θ fK(w),w) , (3.1)where ~θ is the position on the sky, w(z) is the comoving radial distance to redshift z, wH is the distanceto the horizon, and W κ(w) is the lensing kernel (van Waerbeke et al., 2014),W κ(w) =32Ωm(H0c)2g(w)fK(w)a, (3.2)with δm(~θ fK(w),w) representing the 3D mass density contrast, fK(w) is the angular diameter distanceat comoving distance w, and the function g(w) depends on the source redshift distribution n(w) asg(w) =∫ wHwdw′ n(w′)fK(w′−w)fK(w′), (3.3)where we choose the following normalization for n(w):∫ ∞0dw′ n(w′) = 1. (3.4)The tSZ signal is due to the inverse Compton scattering of CMB photons off hot electrons along theline-of-sight which results in a frequency-dependent variation in the CMB temperature (Sunyaev et al.,1970),∆TT0= ySSZ(x), (3.5)52where SSZ(x) = xcoth(x/2)−4 is the tSZ spectral dependence, given in terms of x = hν/kBT0. Here his the Planck constant and should not be confused with the parametrization of the Hubble rate H0, kBis the Boltzmann constant, and T0 = 2.725 K is the CMB temperature. The quantity of interest in thecalculations here is the Comptonization parameter, y, given by the line-of-sight integral of the electronpressure:y(~θ) =∫ wH0adwkBσTmec2neTe, (3.6)where σT is the Thomson cross-section, kB is the Boltzmann constant, and ne[~θ fK(w),w] andTe[~θ fK(w),w] are the 3D electron number density and temperature, respectively.The first estimator of the tSZ-lensing cross-correlation that we use for the analysis in this chapter isthe configuration-space two-point cross-correlation function, ξ y−κ(ϑ):ξ y−κ(ϑ) = ∑`(2`+14pi)Cy−κ` P` (cos(ϑ))by` bκ` , (3.7)where P` are the Legendre polynomials. Note that ϑ represents the angular separation and should notbe confused with the sky coordinate ~θ . The y–κ angular cross-power spectrum isCy−κ` =12`+1∑my`mκ∗`m, (3.8)where y`m and κ`m are the spherical harmonic transforms of the y and κ maps, respectively (see Maet al. 2015 for details), and by` and bκ` are the smoothing kernels of the y and κ maps, respectively.Note that we ignore higher-order lensing corrections to our cross-correlation estimator. It was shown inTro¨ster et al. (2014, Chapter 4) that corrections due to the Born approximation, lens-lens coupling, andhigher-order reduced shear estimations have a negligible contribution to our measurement signal. Wealso ignore relativistic corrections to the tSZ signal.Another estimator of lensing-tSZ correlations is constructed using the tangential shear, γt , which isdefined asγt(~θ) =−γ1 cos(2φ)− γ2 sin(2φ), (3.9)where (γ1,γ2) are the shear components relative to Cartesian coordinates, ~θ = [ϑ cos(φ),ϑ sin(φ)]where φ is the polar angle of ~θ with respect to the coordinate system. In the flat sky approximation, theFourier transform of γt can be written in terms of the Fourier transform of the convergence as (Jeonget al., 2009):γt(~θ) =−∫ d2l(2pi)2κ(~l)cos[2(φ −ϕ)]eilθ cos(φ−ϕ). (3.10)where ϕ is the angle between ~l and the cartesian coordinate system. We use the above expression toderive the y–γt cross-correlation function asξ y−γt (ϑ) = 〈y γt〉(ϑ) =∫ 2pi0dφ2pi∫ d2l(2pi)2Cy−κl cos[2(φ −ϕ)]eilϑ cos(φ−ϕ). (3.11)53Note that the correlation function that we have introduced in Eq. 3.11 differs from what is commonlyused in galaxy-galaxy lensing studies, where the average shear profile of haloes 〈γt〉 is measured. Here,we take every point in the y map, compute the corresponding tangential shear from every galaxy atangular separation ϑ in the shear catalogue and then take the average (instead of computing the signalaround identified haloes). Working with the shear directly in this way, instead of convergence, has theadvantage that we skip the mass map reconstruction process and any noise and systematic issues thatmight be introduced during the process. We have successfully applied similar estimators previously tocompute the cross-correlation of galaxy lensing with CMB lensing in Harnois-De´raps et al. (2016). Inprinciple, this estimator can be used for cross-correlations with any other scalar quantity.Fourier-space versus configuration-space analysisIn addition to the configuration-space analysis described above, we also study the cross-correlationin the Fourier space. A configuration-space analysis has the advantage that there are no complica-tions introduced by the presence of masks, which significantly simplifies the analysis. As described inHarnois-De´raps et al. (2016), a Fourier analysis requires extra considerations to account for the impactof several factors, including the convolution of the mask power spectrum and mode-mixing. On theother hand, a Fourier space analysis can be useful in distinguishing between different physical effects atdifferent scales (e.g., the impact of baryon physics and AGN feedback). We choose a forward modellingapproach as described in Harnois-De´raps et al. (2016) and discussed further in Section Observational dataRCSLenS lensing mapsThe Red Sequence Cluster Lensing Survey (Hildebrandt et al., 2016) is part of the second Red-sequenceCluster Survey (RCS2; Gilbank et al. (2011)).1 Data were acquired from the MegaCAM camera from14 separate fields and covers a total area of 785 deg2 on the sky. The pipeline used to process RCSLenSdata includes a reduction algorithm (Erben et al., 2013), followed by photometric redshift estimation(Hildebrandt et al., 2012; Benı´tez, 2000) and a shape measurement algorithm (Miller et al., 2013). Fora complete description see Heymans et al. (2012) and Hildebrandt et al. (2016).For some of the RCSLenS fields the photometric information is incomplete, so we use external datato estimate the galaxy source redshift distribution, n(z). The CFHTLenS-VIPERS photometric sampleis used which contains near-UV and near-IR data combined with the CFHTLenS photometric sampleand is calibrated against 60’000 spectroscopic redshifts (Coupon et al., 2015). The source redshiftdistribution, n(z), is then obtained by stacking the posterior distribution function of the CFHTLenS-VIPERS galaxies with predefined magnitude cuts and applying the following fitting function (using the1The RCSLenS data are public and can be found at: www.rcslens.org54procedure outlined in Section 3.1.2 of Harnois-De´raps et al. (2016)):nRCSLenS(z) = a z exp[−(z−b)2c2]+d z exp[−(z− e)2f 2]+ g z exp[−(z−h)2i2]. (3.12)As described in the Appendix B.1, we experimented with several different magnitude cuts to find therange where the SNR for our measurements is maximized. We find that selecting galaxies with magr >18 yields the highest SNR with the best-fit values of (a,b,c,d,e, f ,g,h, i) = (2.94, −0.44, 1.03, 1.58,0.40, 0.25, 0.38, 0.81, 0.12). This cut leaves us with approximately 10 million galaxies from the 14RCSLenS fields, yielding an effective galaxy number density of n¯ = 5.8 gal/arcmin2 and an ellipticitydispersion of σε = 0.277 (see Heymans et al. (2012) for details).Fig. 3.1 shows the source redshift distributions n(z) for the three different magnitude cuts we haveexamined. The double-bump structure in n(z) for the 18–24 and > 18 magr selection is due to the in-terplay of the galaxy selection function and the shapes of the posterior redshift distribution function ofthe CFHTLenS-VIPERS galaxies. Note that the lensing signal is most sensitive in the redshift range ap-proximately half way between the sources and the observer. RCSLenS is shallower than the CFHTLenS(see the analysis in van Waerbeke et al., 2013) but, as we demonstrate later, the larger area coverage ofRCSLenS (more than) compensates for the lower number density of the source galaxies, in terms of themeasurement of the cross-correlation with the tSZ signal.0.0 0.5 1.0 1.5 2.0 2.5 3.0z0.> 1818-2421.5-23Figure 3.1: Redshift distribution, n(z), of the RCSLenS sources for different r-magnitude cuts. Wework with the magr > 18 cut (which includes all the objects in the survey).For our analysis we use the shear data as well as the reconstructed projected mass maps (con-vergence maps) from RCSLenS. For the tSZ-tangential shear cross-correlation (y–γt), we work at the55catalogue level where each pixel in the y map is correlated with the average tangential shear from thecorresponding shear data in an annular bin around that point, as described in Section 3.3.1. To constructthe convergence maps, we follow the method described in van Waerbeke et al. (2013). In AppendixB.1 we study the impact of map smoothing on the signal to noise ratio (SNR) we determine for the y–κcross-correlation analysis. We demonstrate that the best SNR is obtained when the maps are smoothedwith a kernel that roughly matches the beam scale of the corresponding y maps from Planck survey(FWHM = 10 arcmin).The noise properties of the constructed maps are studied in detail in Appendix B.2.Planck tSZ y mapsFor the cross-correlation with the tSZ signal, we use the full sky maps provided in the Planck 2015 publicdata release (Planck Collaboration XXII, 2016). We use the MILCA map that has been constructedfrom multiple frequency channels of the survey. Since we are using the public data from the Planckcollaboration, there is no significant processing involved. Our map preparation procedure is limited tomasking the map and cutting the patches matching the RCSLenS footprint.Note that in performing the cross-correlations we are limited by the footprint area of the lensingsurveys. In the case of RCSLenS, we have 14 separate compact patches with different sizes. In contrast,the tSZ y maps are full-sky (except for masked regions). We therefore have the flexibility to cut out largerregions around the RCSLenS fields, in order to provide a larger cross-correlation area that helps suppressthe statistical noise, leading to an improvement in the SNR. We cut out y maps so that there is completeoverlap with RCSLenS up to the largest angular separation in our cross-correlation measurements.Templates have also been released by the Planck collaboration to remove various contaminatingsources. We use their templates to mask Galactic emission and point sources, which amounts to remov-ing about 40% of the sky. We have compared our cross-correlation measurements with and without thetemplates and checked that our signal is robust. We have also separately checked that the masking ofpoint sources has a negligible impact on our cross-correlation signal (see Appendix B.1). These sourcesof contamination do not bias our cross-correlation signal and contribute only to the noise level.In addition to using the tSZ map from the Planck collaboration, we have also tested our cross-correlation results with the maps made independently following the procedure described in van Waer-beke et al. (2014), where several full-sky y maps were constructed from the second release of PlanckCMB maps. To construct the maps, a linear combination of the four HFI frequency band maps (100,143, 217, and 353 GHz) were used and smoothed with a Gaussian beam profile with θSZ,FWHM = 10arcmin. To combine the band maps, band coefficients were chosen such that the primary CMB signal isremoved, and the dust emission with a spectral index βd is nulled. A range of models with different βdvalues were employed to construct a set of y maps that were used as diagnostics of residual contamina-tion. The resulting cross-correlation measurements vary by roughly 10% between the different y maps,but are consistent within the errors with the measurements from the public Planck map.563.2.3 Theoretical modelsWe compare our measurements with theoretical predictions based on the halo model and from fullcosmological hydrodynamical simulations. Below we describe the important aspects of these models.Halo modelWe use the halo model description for the tSZ - lensing cross-correlation developed in Ma et al. (2015).In the framework of the halo model as introduced in Chapter 1, the y–κ cross-correlation power spectrumis:Cy−κ` =Cy−κ,1h` +Cy−κ,2h` , (3.13)where the 1-halo and 2-halo terms are defined asCy−κ,1h` =∫ zmax0dzdVdzdΩ∫ MmaxMmindMdndMy`(M,z)κ`(M,z),Cy−κ,2h` =∫ zmax0dzdVdzdΩPlinm (k = `/χ,z)×[∫ MmaxMmindMdndMb(M,z)κ`(M,z)]×[∫ MmaxMmindMdndMb(M,z)y`(M,z)]. (3.14)In the above equations Plinm (k,z) is the 3D linear matter power spectrum at redshift z, κ`(M,z) is theFourier transform of the convergence profile of a single halo of mass M at redshift z with the NFWprofile:κ` =W κ(z)χ2(z)1ρ¯m∫ rvir0dr(4pir2)sin(`r/χ)`r/χρ(r;M,z), (3.15)and y`(M,z) is the Fourier transform of the projected gas pressure profile of a single halo:y` =4pirs`2sσTmec2∫ ∞0dxx2sin(`x/`s)`x/`sPe(x;M,z). (3.16)Here x≡ a(z)r/rs and `s = aχ/rs, where rs is the scale radius of the 3D pressure profile, and Pe is the 3Delectron pressure. The ratio rvir/rs is the concentration parameter (see e.g Ma et al. (2015) for details).For the electron pressure of the gas in haloes, we adopt the so-called ‘universal pressure profile’(UPP; Arnaud et al. 2010):P(x≡ r/R500) = 1.65×10−3E(z) 83(M5003×1014h−170 M) 23+0.12× P(x)h270[keV cm−3], (3.17)57Table 3.1: Sub-grid physics of the baryon feedback models in the cosmo-OWLS runs. Each modelhas been run adopting both the WMAP-7yr and Planck cosmologies.Simulation UV/X-ray Cooling Star SN AGN ∆Theatbackground formation feedback feedbackNOCOOL Yes No No No No ...REF Yes Yes Yes Yes No ...AGN 8.0 Yes Yes Yes Yes Yes 108.0 KAGN 8.5 Yes Yes Yes Yes Yes 108.5 KAGN 8.7 Yes Yes Yes Yes Yes 108.7 Kwhere P(x) is the generalized NFW model (Nagai et al., 2007):P(x) =P0(c500x)γ [1+(c500x)α ](β−γ)/α . (3.18)We use the best-fit parameter values from Planck Collaboration Int. V (2013): {P0,c500,α,β ,γ} ={6.41,1.81,1.33,4.13,0.31}. Since we ultimately only fit for a single amplitude parameter AtSZ (de-fined in Eq. 3.24), we choose not to marginalise over the individual UPP parameters but instead fix themto the values provided in Planck Collaboration Int. V (2013). To compute the configuration-space cor-relation functions, we use Eqs. 3.7 and 3.11 for ξ y−κ and ξ y−γt , respectively. We present the halo modelpredictions for two sets of cosmological parameters: the maximum-likelihood Planck 2013 cosmol-ogy (Planck Collaboration XVI, 2014) and the maximum-likelihood WMAP-7yr cosmology (Komatsuet al., 2011) with {Ωm, Ωb, ΩΛ, σ8, ns, h} = 0.3175, 0.0490, 0.6825, 0.834, 0.9624, 0.6711 and {0.272,0.0455, 0.728, 0.81, 0.967, 0.704}, respectively.There are several factors that have an impact on these predictions; the choice of the gas pressureprofile, the adopted cosmological parameters, and the n(z) distribution of sources in the lensing survey.In addition, the hydrostatic mass bias parameter, b (defined as Mobs,500 = (1− b)Mtrue,500), alters therelation between the adopted pressure profile and the true halo mass. Typically, it has been suggestedthat 1− b ≈ 0.8. Note that the impact of the hydrostatic mass bias in real groups and clusters will beabsorbed into our amplitude fitting parameter AtSZ.3.2.4 The cosmo-OWLS hydrodynamical simulationsWe also compare our measurements to predictions from the cosmo-OWLS suite of hydrodynamicalsimulations. In Hojjati et al. (2015) we compared these simulations to measurements using CFHTLenSdata and we also demonstrated that high resolution tSZ-lensing cross-correlations have the potential tosimultaneously constrain cosmological parameters and baryon physics. Here we build on our previouswork and employ the cosmo-OWLS simulations in the modelling of RCSLenS data.The cosmo-OWLS suite is an extension of the OverWhelmingly Large Simulations project (OWLS;Schaye et al. 2010). The suite consists of box-periodic hydrodynamical simulations with volumes of(400 h−1 Mpc)3 and 10243 baryon and dark matter particles. The initial conditions are based on either58the WMAP-7yr or Planck 2013 cosmologies. We quantify the agreement of our measurements with thepredictions from each cosmology in Section 3.5 .We use five different baryon models from the suite as summarized in Table 3.1 and described in detailin Le Brun et al. (2014) and McCarthy et al. (2014) and references therein. NOCOOL is a standard non-radiative (‘adiabatic’) model. REF is the OWLS reference model and includes sub-grid prescriptions forstar formation (Schaye et al., 2008), metal-dependent radiative cooling (Wiersma et al., 2009a), stellarevolution, mass loss, chemical enrichment (Wiersma et al., 2009b), and a kinetic supernova feedbackprescription (Dalla Vecchia et al., 2008). The AGN models are built on the REF model and additionallyinclude a prescription for black hole growth and feedback from active galactic nuclei (Booth et al.,2009). The three AGN models differ only in their choice of the key parameter of the AGN feedbackmodel ∆Theat, which is the temperature by which neighbouring gas is raised due to feedback. Increasingthe value of ∆Theat results in more energetic feedback events, and also leads to more bursty feedback,since the black holes must accrete more matter in order to heat neighbouring gas to a higher adiabat.Following McCarthy et al. (2014), we produce light cones of the simulations by stacking randomlyrotated and translated simulation snapshots (redshift slices) along the line-of-sight back to z = 3. Notethat we use 15 snapshots at fixed redshift intervals between z= 0 and z= 3 in the construction of the lightcones. This ensures a good comoving distance resolution, which is required to capture the evolution ofthe halo mass function and the tSZ signal. The light cones are used to produce 5◦× 5◦ maps of they, shear (γ1, γ2) and convergence (κ) fields. We construct 10 different light cone realizations for eachfeedback model and for the two background cosmologies. Note that in the production of the lensingmaps we adopt the source redshift distribution, n(z), from the RCSLenS survey to produce a consistentcomparison with the observations.From our previous comparisons to the cross-correlation of CFHTLenS weak lensing data with theinitial public Planck data in Hojjati et al. (2015), we found that the data mildly preferred a WMAP-7yrcosmology to the Planck 2013 cosmology. We will revisit this in Section 3.5 in the context of the newRCSLenS data.3.3 Observed cross-correlationBelow we describe our cross-correlation measurements between tSZ y and galaxy lensing quantitiesusing the configuration-space and Fourier-space estimators described in Section Configuration-space analysisWe perform the cross-correlations on the 14 RCSLenS fields. The measurements from the fields con-verge around the mean values at each bin of angular separation with a scatter that is due to statisticalnoise and sampling variance. To combine the fields, we take the weighted mean of the field measure-ments, where the weights are determined by the total LENSFIT weight (see Miller et al. 2013 for technicaldefinitions).As described earlier, to improve the SNR and suppress statistical noise, we use ‘extended’ y mapsaround each RCSLenS field to increase the cross-correlation area. For RCSLenS, we extend our mea-5920 40 60 80 100 120 140 160 180ϑ [arcmin]−ξy−κ(ϑ)×10−9PlanckWMAP20 40 60 80 100 120 140 160 180ϑ [arcmin]ξy−γt(ϑ)×10−9PlanckWMAPFigure 3.2: Cross-correlation measurements of y–κ (left) and y–γt (right) from RCSLenS. Thelarger (smaller) error bars represent uncertainties after (before) including our estimate of thesampling variance contribution (see Section 3.4). Halo model predictions using UPP withWMAP-7yr and Planck cosmologies are also over-plotted for comparison.surements to an angular separation of 3◦, and hence include 4◦ wide bands around the RCSLenS fields.We compute our configuration-space estimators as described below. For y–γt , we work at the cata-logue level and compute the two-point correlation function asξ y−γt (ϑ) =∑i j yiei jt w j∆i j(ϑ)∑i j w j∆i j(ϑ)11+K(ϑ), (3.19)where yi is the value of pixel i of the tSZ map, ei jt is the tangential ellipticity of galaxy j in the cata-log with respect to pixel i, and w j is the LENSFIT weight. The (1+K(ϑ))−1 factor accounts for themultiplicative calibration correction (see Hildebrandt et al. (2016) for details):11+K(ϑ)=∑i j w j∆i j(ϑ)∑i j w j(1+m j)∆i j(ϑ). (3.20)Finally, ∆i j(ϑ) is imposes our binning scheme and is 1 if the angular separation is inside the bin centredat ϑ and zero otherwise.For the y–κ cross-correlation, we use the corresponding mass maps for each field and compute thecorrelation function asξ y−κ(ϑ) =∑i j yiκ j∆i j(ϑ)∑i j∆i j(ϑ), (3.21)where κ j is the convergence value at pixel j and includes the necessary weighting, w j.Fig. 3.2 presents our configuration-space measurement of the RCSLenS cross-correlation withPlanck tSZ. Our measurements are performed within 8 bins of angular separation, square-root-spacedbetween 1 and 180 arcmin. That is, the bins are uniformly spaced between√1′ and√180′. The filledcircle data points show the y–κ (left) and y–γt (right) cross-correlations. To guide the eye, the solid60red curves and dashed green curves represent the predictions of the halo model for WMAP-7yr andPlanck 2013 cosmologies, respectively.3.3.2 Fourier-space measurements0 500 1000 1500 2000`−1012345678`3Cy−κ`×10−6PlanckWMAPFigure 3.3: Similar to Fig. 3.2 but for Fourier-space estimator, Cy−κ` .In the Fourier-space analysis, we work with the convergence and tSZ maps. As detailed in Harnois-De´raps et al. (2016), it is important to account for a number of numerical and observational effects whenperforming the Fourier-space analysis. These effects include data binning, map smoothing, masking,zero-padding and apodization. Failing to take such effects into account will bias the cross-correlationmeasurements significantly.Here we adopt the forward modelling approach described in Harnois-De´raps et al. (2016), wheretheoretical predictions are turned into ‘pseudo-C`’, as summarized below. First, we obtain the theo-retical C` predictions from Eqs. (3.13) and (3.14) as described in Section 3.2.3. We then multiply thepredictions by a Gaussian smoothing kernel that matches the Gaussian filter used in constructing the κmaps in the mass map making process, and another smoothing kernel that accounts for the beam effectof the Planck satellite.Next we include the effects of observational masks on our power spectra which breaks down intothree components (see Harnois-De´raps et al. (2016) for details): i) an overall downward shift of powerdue to the masked pixels which can be corrected for with a rescaling by the number of masked pixels;ii) an optional apodization scheme that we apply to the masks to smooth the sharp features introducedin the power spectrum of the masked map that enhance the high-` power spectrum measurements; andiii) a mode mixing matrix, that propagates the effect of mode coupling due to the observational window.As shown in Harnois-De´raps et al. (2016), steps (ii) and (iii) are not always necessary in the contextof cross-correlation when the masks from both maps do not strongly correlate with the data. We havechecked that this is indeed the case by measuring the cross-correlation signal from the cosmo-OWLSsimulations with and without applying different sections of the RCSLenS masks, with and withoutapodization, and observed that changes in the results were minor. We therefore choose to remove the61steps (ii) and (iii) from the analysis pipeline. As the last step in our forward modelling, we re-bin themodelled pseudo-C` so that it matches the binning scheme of the data. Note that these steps have to becalculated separately for each individual field due to their distinct masks.Fig. 3.3 shows our Fourier-space measurement for the y–κ cross-correlation, where halo model pre-dictions for the WMAP-7yr and Planck cosmologies are also over-plotted. Our Fourier-space measure-ment is consistent with the configuration-space measurement overall. Namely, the data points provide abetter match to WMAP-7yr cosmology prediction on small physical scales (large ` modes) and tend tomove towards the Planck prediction on large physical scales (small ` modes). A more detailed compar-ison is non-trivial as different scales (` modes) are mixed in the configuration-space measurements.The details of error estimation and the significance of the detection are described in Section Estimation of covariance matrices and significance of detectionIn this section, we describe the procedure for constructing the covariance matrix and the statisticalanalysis that we perform to estimate the significance of our measurements. We have investigated severalmethods for estimating the covariance matrix for the type of cross-correlations performed in this chapter.3.4.1 Configuration-space covarianceTo estimate the covariance matrix we follow the method of van Waerbeke et al. (2013). We first produce300 random shear catalogues from each of the RCSLenS fields. We create these catalogs by randomlyrotating the individual galaxies. This procedure will destroy the underlying lensing signal and createcatalogs with pure statistical lensing noise. We then construct the y–γt covariance matrix, Cy−γt , bycross-correlating the randomized shear maps for each field with the y map.To construct the y–κ covariance matrix, we perform our standard mass reconstruction procedure oneach of the 300 random shear catalogs to get a set of convergence noise maps. We then compute thecovariance matrix by cross-correlating the y maps with these random convergence maps. We follow thesame procedure of map making (masking, smoothing, etc.) in the measurements from random mapsas we did for the actual measurement. This ensures that our error estimation is representative of theunderlying covariance matrix.Note that we also need to ‘debias’ the inverse covariance matrix by a debiasing factor as describedin Hartlap et al. (2007):α = (n− p−2)/(n−1) , (3.22)where p is the number of data bins and n is the number of random maps used in the covariance estima-tion2.The correlation coefficients are shown in Fig. 3.4 for y–κ (left) and y–γt (right). As a characteristicof configuration-space, there is a high level of correlation between pairs of data points within eachestimator. This is more pronounced for y–κ since the mass map construction is a non-local operation,2In principle we should also implement the treatment of Sellentin et al. (2016), but the precision of our measurement is nothigh enough to worry about such errors.62and also that the maps are smoothed which creates correlation by definition. Having a lower level of bin-to-bin correlations is another reason why one might want to work with tangential shear measurementsrather than mass maps in such cross-correlation studies.4 25 65 124ϑ [arcmin]42565124ϑ[arcmin] 25 65 124ϑ [arcmin]42565124ϑ[arcmin] 670 105014301810`290670105014301810`− 3.4: Cross-correlation coefficient matrix of the angular bins for the configuration-spacey−κ (left) and y–γt (middle), and the Fourier-space y−κ (right) estimators. Angular binsare more correlated for the y–κ estimator compared to y–γt or the Fourier-space estimator.3.4.2 Fourier-space covarianceFor the covariance matrix estimation in Fourier space, we follow a similar procedure as in configurationspace. We first Fourier transform the random convergence maps, and then follow the same analysis forthe measurements (see Section 3.3). The resulting cross-correlation measurements create a large samplethat can be used to construct the covariance matrix. Similar to the configuration space analysis, we alsodebias the computed covariance matrix.Figure 3.4, right shows the cross-correlation coefficients for the ` bins (Note that we chose to workwith 5 linearly-spaced bins between ` = 100 and ` = 2000). As expected, there is not much bin-to-bincorrelation and the off-diagonal elements are small.3.4.3 Estimating the contribution from the sampling varianceConstructing the covariance matrix as described above includes the statistical noise contribution only.There is, however, a considerable scatter in the cross-correlation signal between the individual fields. Acomparison of the observed scatter to that among different lines of sight (LoS) of the (noise-free) sim-ulations shows that the sampling variance contribution is non-negligible. We therefore need to includethe contribution to the covariance matrix from sampling variance.We are not able, however, to estimate a reliable covariance matrix that includes sampling variancesince the number of samples we have access to is very limited and the resulting covariance matrix will benoisy and non-invertible. We only have a small number of fields from the lensing surveys (14 fields fromRCSLenS is not nearly enough) and the same is true for the number of LoS maps from hydrodynamical630 36 72 108 144 180ϑ [arcmin]0246810121416Cfieldii/Cstatii0 36 72 108 144 180ϑ [arcmin]123456789Cfieldii/Cstatii290 670 1050 1430 1810`0123456789Cfieldii/CstatiiFigure 3.5: Ratios of the variance between the 14 RCSLenS fields and the variance estimated fromrandom shear maps, as described in Sec. 3.4. The best-fitting linear model for the ratios areshown in green.simulations (10 LoS). Instead, we can estimate the sampling variance contribution by quantifying byhow much we need to ‘inflate’ our errors to account for the impact of sampling variance.Note that the scatter in the cross-correlation signal from the individual fields is due to both statisticalnoise and sampling variance. We compare the scatter (or variance) in each angular bin to that of thediagonal elements of the reconstructed covariance matrix that we obtained from the previous section(which quantifies the statistical uncertainty alone). We estimate the scaling factor by which we shouldinflate the computed covariance matrix to match the observed scatter.The ratio between the variance between the fields and the statistical covariance is scale-independentfor the tangential shear but shows some scale dependence for the other two estimators. We thereforewish to find a simple description of the ratio ri at some scale i between the field variance and statisticalcovariance, such that the statistical covariance Cstat can be rescaled asCi j = Cstati j√rir j . (3.23)We model the ratio ri as a liner function of the scale. The model is then fit to the observer ratios betweenthe field variances and statistical covariance. The errors on the observed ratios are estimated by taking1000 bootstrap resamplings of the 14 RCSLenS fields and calculating the ratio from the variance of thoseresampled fields. The errors are highly correlated themselves but the error covariance is not invertiblefor the same reason the data covariance of the 14 fields is not invertible; the number of independent fieldsis too small. The observed ratios and the best-fitting models ri are shown on Fig. 3.5. We use these best-fitting models to rescale the statistical covariance according to Eq. (3.23) to obtained an estimate of thefull covariance.3.4.4 χ2 analysis and significance of detectionWe quantify the significance of our measurements using the SNR estimator as described below. Weassume that the RCSLenS fields are sufficiently separated such that they can be treated as independent,ignoring field-to-field covariance.64Estimator DoF χ2null, stat. err. only χ2null, adjustedξ y−κ 8 193.5 56.2ξ y−γt 8 307.4 71.6combined 16 328.7 124.2Cy−κ` 5 156.4 64.9Table 3.2: Summary of the χ2null values before and after including the sampling variance contri-bution according to the adjustment procedure of Section 3.4.3. There are 8 angular bins, ordegrees of freedom (DoF), at which the individual estimators are computed. Combining theestimators increases the DoF accordingly.First, we introduce the cross-correlation bias factor, AtSZ, through:V = ξ˜ −AtSZ ξˆ . (3.24)AtSZ quantifies the difference in amplitude between the measured (ξ˜ ) and predicted (ξˆ ) cross-correlationfunction. The prediction can be from either the halo model or from hydrodynamical simulations. UsingV , we define the χ2 asχ2 = V C−1V T , (3.25)where C is the covariance matrix.We define χ2null by setting AtSZ = 0. In addition, χ2min is found by minimizing Eq. (3.25) with respectto AtSZ:χ2null : AtSZ = 0 ; (3.26)χ2min : AtSZ,min . (3.27)In other words, χ2min quantifies the goodness of fit between the measurements and our model predictionafter marginalizing over AtSZ.Table 3.2 summarizes the χ2null values from the measurements before and after including the sam-pling variance contribution. The values are quoted for individual estimators as well as when they arecombined. The χ2null is always higher for y–γt estimator, demonstrating that it is a better estimator forour cross-correlation analysis. It also improves when we combine the estimators but we should considerthat χ2null increases at the expense of adding extra degrees of freedom. Namely, we have eight angularbins for each estimators and combining the two, there are 16 degrees of freedom which introduces aredundancy due to the correlation between the two estimators so that χ2null does not increase by a factorof 2.Finally, we define the SNR as follows. We wish to quantify how strongly we can reject the nullhypothesis H0, that no correlation exists between lensing and tSZ, in favour of the alternative hypothesisH1, that the cross-correlation is well described by our fiducial model up to a scaling by the cross-65correlation bias AtSZ. To this end, we employ a likelihood ratio method. The deviance D is given by thelogarithm of the likelihood ratio between H0 and H1:D =−2logL (~d|H0)L (~d|H1). (3.28)For Gaussian likelihoods, the deviance can then be written asD = χ2null−χ2min . (3.29)If H1 can be characterized by a single, linear parameter, D is distributed as χ2 with one degree offreedom Williams (2001). The significance in units of standard deviations σ of the rejection of the nullhypothesis, i.e., the significance of detection, is therefore given by:SNR =√χ2null−χ2min. (3.30)Table 3.3 summarizes the significance analysis of our measurements. We show the SNR and best-fitamplitude AtSZ, for the theoretical halo model predictions with WMAP-7yr and Planck cosmologies.The results are presented for each estimator independently as well as for their combination. Note thatall the values in Table 3.3 are adjusted to account for the sampling variance, as described in Section3.4.3. To estimate the combined covariance matrix, we place the covariance for individual estimatorsas block diagonal elements of combined matrix and compute the off-diagonal blocks (the covariancebetween the two estimators).The predictions from WMAP-7yr cosmology are relatively favored in our analysis, which is consis-tent with the results of previous studies (e.g. McCarthy et al. 2014; Hojjati et al. 2015). We, however,find similar SNR values from both cosmologies because the effect of the different cosmologies on thehalo model prediction can be largely accounted for by an overall rescaling (AtSZ). After rescaling, theremaining minor differences are due to the shape of the cross-correlation signal so that the SNR dependsonly weakly on the cosmology.We obtain a 13.3σ and 16.8σ from y–κ and y–γt estimators, respectively, when we only considerthe statistical noise in the covariance matrix (before the adjustment prescription of Section 3.4.3) 3.The 13.3σ significance from y–κ estimator should be compared to the ∼6σ detection from thesame estimator in van Waerbeke et al. (2014) where CFHTLenS data are used instead. As expected,RCSLenS yields an improvement in the SNR and y–γt improves it further. Including sampling variancein the covariance matrix decreases the detection significance from RCSLenS data to 7.1σ and 8.1σ forthe y–κ and y–γt estimators, respectively.We perform a similar analysis in Fourier space where the data vector is given by the pseudo-C`s andthe results are included in Table 3.3. The SNR values are in agreement with the configurations-space3Note that we are quoting the detection levels from a statistical noise-only covariance matrix so that we can compare to theprevious literature, including the results of van Waerbeke et al. (2014), where a similar approach is taken in the constructionof the covariance matrix (i.e. only statistical noise is considered).66Estimator SNR stat. err. only SNR adjusted Atsz, WMAP-7yr Atsz, Planckξ y−κ 13.3 7.1 1.18±0.17 0.64±0.09ξ y−γt 16.8 8.1 1.27±0.16 0.68±0.08combined 17.1 10.6 1.23±0.12 0.66±0.06Cy−κ` 11.6 7.5 1.07±0.14 0.60±0.08Table 3.3: Summary of the statistical analysis of the cross-correlation measurements. For theconfiguration-space estimators, the results are shown for each estimator independently andwhen they are combined. SNR quantifies the significance of detection after a fit to modelpredictions (halo model). SNR values are shown before (SNR, stat. err. only) and after (SNR,adjusted) adjustment for sampling variance uncertainties according to the description of Sec-tion 3.4.3, while AtSZ values are quoted after the adjustment. The Planck cosmology predictshigher amplitude than WMAP-7yr cosmology so that overall, the WMAP-7yr cosmology pre-dictions are in better agreement with the measurements.analysis4. We see a similar trend as in the configuration-space analysis in that there is a better agreementwith the WMAP-7yr halo model predictions (AtSZ is closer to 1) while the Planck cosmology predicts ahigh amplitude.Table 3.3 summarizes the predictions from the halo model framework with a fixed pressure profilefor gas (UPP). In Section 3.5, we revisit this by comparing to predictions from hydrodynamical simula-tions where haloes with different mass and at different redshifts have a variety of gas pressure profiles.We show that we find better agreement with models where AGN feedback is present in haloes.Impact of maximum angular separationThe two configuration-space estimators we use probe different dynamical scales by definition. Thismeans that as we include cross-correlations at larger angular scales, information is captured at a differentrate by the two estimators. For a survey with limited sky coverage, combining the two estimators willtherefore improve the SNR of the measurements. In the following, we quantify this improvement of theSNR.In Fig. 3.6, we plot the SNR values as a function of the maximum angular separation for bothestimators. In addition, we also include the same quantities when debiased using Eq.3.22 to highlightthe effect of increasing DoF on the debiasing factor. Each angular bin is 10 arcmin wide and addingmore bins means including cross-correlation at larger angular scales.We observe that the y–κ measurement starts off with a higher SNR relative to y–γt at small angularseparations. The SNR in y–κ levels off very quickly with little information added above 1◦ separation.The shallow SNR slope of the y–κ curve is partly due to the Gaussian smoothing kernel that is used inreconstructing the mass maps which spreads the signal within the width of the kernel. The y–γt cross-correlation, on the other hand, has a higher rate of gain in SNR and catches up with the convergencerapidly. Eventually, the two estimators approach a plateau as the cross correlation signals drops to zero.4Note that the Fourier-space analysis is performed by pipeline 3 in Harnois-De´raps et al. (2016). Different pipelines giveslightly different but consistent results.67At that point, both contain the same amount of cross-correlation information.Note that we limit ourselves to a maximum angular separation of 3 degrees in the RCSLenS mea-surements since the measurement is very noisy beyond that. Fig. 3.6 indicates that the two estimatorsmight not have converged to the limit where the information is saturated (a plateau in the SNR curve).Since each estimator captures different information up to 3 degrees, combining them improves the mea-surement significance (see Table 3.3). With surveys like the Kilo-Degree Surveys (KiDS) de Jonget al. (2013a), the Dark Energy Survey (DES) Abbott et al. (2016) and the Hyper Suprime-Cam Survey(HSC) Miyazaki et al. (2012), where the coverage area is larger, we will be able to go to larger angularseparations where the information from our estimators is saturated. The signal at such large scales isprimarily dependent on cosmology and quite independent of the details of the astrophysical processesinside haloes (see Hojjati et al. 2015 for more details). This could, in principle, provide a new probe ofcosmology based on the cross-correlation of baryons and lensing on distinct scales and redshifts.20 40 60 80 100 120 140 160 180< ϑ [arcmin]10121416182022SNR(<ϑ)y − κy − κ debiasedy − γty − γt debiasedFigure 3.6: SNR as a function of the maximum angular separation for the two configuration-spaceestimators, with and without debiasing (as described in Sec. 3.4.1). The slope of the linesis different for the two estimators due to the different information they capture as a functionof angular separation. For y–κ , most of the signal is in the first few bins making it a bettercandidate at small scales, while y–γt has a larger slope and catches up quickly. Eventually, thetwo estimators approach a plateau where they contain the same amount of cross-correlationinformation. Note that the plots show the SNR before the ‘adjustment’ procedure of Sec. Implications for cosmology and astrophysicsIn Section 3.3 we compared our measurements to predictions from the theoretical halo model. Thehalo model approach, however, has limitations for the type of cross-correlation we are considering. Forexample, in our analysis we cross-correlate every source in the sky that produces a tSZ signal withevery source that produces a lensing signal. A fundamental assumption of the halo model, however,68is that all the mass in the Universe is in spherical haloes, which is not an accurate description of thelarge-scale structure in the Universe. There are other structures such as filaments, walls or free flowingdiffuse gas in the Universe, so that matching them with spherical haloes could lead to biased inferenceof results (see Hojjati et al. 2015). Another shortcoming is that our halo model analysis considers a fixedpressure profile for diffuse gas, while it has been demonstrated in various studies that the UPP does notnecessarily describe the gas around low-mass haloes particularly well (e.g., Battaglia et al. 2012; LeBrun et al. 2015).Here we employ the cosmo-OWLS suite of cosmological hydrodynamical simulations (see Section3.2.4) which includes various simulations with different baryonic feedback models. The cosmo-OWLSsuite provides a wide range of tSZ and lensing (convergence and shear) maps allowing us to study theimpact of baryons on the cross-correlation signal. We follow the same steps as we did with the real datato extract the cross-correlation signal from the simulated maps.Figure 3.7 compares our measured configuration-space cross-correlation signal to those from simu-lations with different feedback models. The plots are for the five baryon models using the WMAP-7yrcosmology and the AGN 8.0 model using the Planck cosmology is also plotted for comparison. Notethat baryon models make the largest difference at small scales due to mechanisms that change the den-sity and temperature of the gas inside clusters. For the (non-physical) NOCOOL model, the gas canreach very high densities near the centre of dark matter haloes and is very hot since there is no coolingmechanism in place. This leads to a high tSZ and hence a high cross-correlation signal. After includingthe main baryonic processes in the simulation (e.g. radiative cooling, star formation, SN winds), wesee that the signal drops on small scales. Adding AGN feedback warms up the gas but also expels it tolarger distances from the centre of haloes. This explains why we see a lower signal at small scales buta higher signal at intermediate scales for the AGN 8.7 model. Note that the scatter of the LoS signalvaries for different models due to the details of the baryon processes so that, for example, the AGN 8.7model creates a larger sampling variance. The mean signal of the feedback models is also affected bythe cosmological parameters at all scales. Adopting a Planck cosmology produces a higher signal at allscales and for all models. This is mainly due to the larger values of Ωb, ΩM, and particularly σ8, in thePlanck 2013 cosmology compared to that of the WMAP-7yr cosmology.We summarize in Table 3.4 our χ2 analysis for feedback model predictions relative to our measure-ments. We find that the data prefers a WMAP-7yr cosmology to the Planck 2013 cosmology for all ofthe baryon feedback models. The best-fit models are underlined for both cosmologies.Our measurements are limited by the relatively low resolution of the Planck tSZ maps. On smallscales, our signal is diluted due to the convolution of the tSZ maps with the Planck beam (FWHM= 10 arcmin) which makes it hard to discriminate the feedback models with our configuration-spacemeasurements. This highlights that high-resolution tSZ measurements can be particularly useful toovercome this limitation and open up the opportunity to discriminate the feedback models from tSZ-lensing cross-correlations.The feedback models could be better discriminated in Fourier-space through their power spectra. Wetherefore repeat our measurements on the simulation in the Fourier-space. We apply the same procedure6920 40 60 80 100 120 140 160 180ϑ [arcmin]ξy−γt(ϑ)×10−9AGN 8.0 PlanckAGN 8.0AGN 8.5AGN 8.7REFNOCOOLFigure 3.7: Comparisons of our cross-correlation measurement from RCSLenS to predictionsfrom hydrodynamical simulations. The larger (smaller) error bars represent uncertaintiesafter (before) including our estimate of the sampling variance contribution (see Section 3.4).Different baryon feedback models with WMAP-7yr cosmology are shown for y–γt estimator(we are not showing the plots for y–κ as they are very similar). Baryon feedback has animpact on the cross-correlation signal at small scales.described in Section 3.3.2 to the simulated maps. Fig. 3.8 compares our power spectrum measurementsto simulation predictions, and we have summarized the χ2 analysis in Table 3.4.0 500 1000 1500 2000`−1012345678`3Cy−κ`×10−6AGN 8.7PlanckAGN 8.0AGN 8.5AGN 8.7NOCOOLREFFigure 3.8: Same as Fig. 3.7 for the Fourier-space estimator, Cy−κ` .Our Fourier-space analysis follows a similar trend as the configuration-space analysis. Namely,there is a general over-prediction of the amplitude, which is worse for the Planck cosmology. For the70Model χ2min WMAP-7yr Atsz WMAP-7yr χ2min Planck Atsz PlanckAGN 8.0 5.3 1.00±0.12 5.4 0.68±0.08AGN 8.5 3.2 1.10±0.13 5.9 0.74±0.09Config. Space AGN 8.7 8.6 1.01±0.13 8.3 0.72±0.09NOCOOL 5.7 0.89±0.11 6.1 0.62±0.08REF 5.7 1.06±0.13 5.8 0.74±0.09AGN 8.0 6.1 0.83±0.11 6.5 0.57±0.07AGN 8.5 3.3 1.01±0.13 4.0 0.68±0.09Fourier Space AGN 8.7 1.4 1.20±0.15 1.7 0.83±0.10NOCOOL 8.7 0.72±0.10 8.2 0.50±0.07REF 7.2 0.86±0.11 7.2 0.61±0.08Table 3.4: Summary of χ2min analysis of the cross-correlation measurements from hydrodynamicalsimulations. The error on the best-fit amplitudes are adjusted according to description ofSection 3.4.3. The best-fit models are underlined for both cosmologies. The WMAP-7yrpredictions fit data better for all baryon models. The AGN 8.5 model is preferred by theconfiguration space measurements while the Fourier space measurements prefer the AGN 8.7model.best-fit WMAP-7yr models the fitted amplitude is consistent with 1. Our estimator prefers the AGN 8.7model in all cases with a Planck cosmology that is different than configuration-space estimator wherethe AGN 8.5 was preferred.5 There are a few reasons for these minor differences. For example, thebinning schemes are different and give different weights to bins of angular separation. Furthermore, theFourier-based result can change by small amounts depending on precisely which pipeline from Harnois-De´raps et al. (2016) is adopted.When we fit the amplitude AtSZ to obtain the χ2min or SNR values, we are in fact factoring out thescaling of the model prediction and are left with the prediction of the shape of the cross-correlationsignal. The shape depends on both the cosmological parameters (weakly) and details of the baryonmodel so that the values in Tables 3.3 and 3.4 are a measure of how well the model shape matchesthe measurement. By comparing the χ2min values from the two cosmologies in Table 3.4, the generalconclusion of our analysis is that significant AGN feedback in baryon models is required to matchthe measurements well. Note, however, that while there are differences in the amplitude (and shape)of the cross-correlation signal, the predictions from the two cosmology are not in tension with themeasurements when one considers the current uncertainties in the values of the cosmological parameters(e.g., σ8).3.6 Summary and discussionWe have performed cross-correlations of the public Planck 2015 tSZ map with the public weak lensingshear data from the RCSLenS survey. We have demonstrated that such cross-correlation measurements5We tested that if large-scale correlations (ϑ > 100 arcmin), where uncertainties are large, are removed from the analysis,the AGN 8.7 model is also equally preferred by the configuration-space estimators.71between two independent data sets are free from contamination by residual systematics in each dataset, allowing us to make an initial assessment of the implications of the measured cross-correlations forcosmology and ICM physics.Our cross-correlations are performed at the map level where every object is contributing to thesignal. In other words, this is not a stacking analysis where measurements are done around identifiedhaloes. Instead, we are probing all the structure in the Universe (haloes, filaments etc) and the associatedbaryon distribution including diffuse gas in the intergalactic medium.We performed our analysis using two configuration-space estimators, ξ y−γt and ξ y−κ , and a Fourier-space analysis with Cy−κ` for completeness. Configuration-space estimators have the advantage that theyare less affected by the details of map making processes (masks, appodization etc) and the analysis isstraightforward. We showed that the estimators probe different dynamical scales so that combining themcan improve the SNR of the measurement.Based only on the estimation of statistical uncertainties, the cross-correlation using RCSLenS data isdetected with a significance of 13.3σ and 16.8σ from the y–κ and y–γt estimators, respectively. Includ-ing a heuristic estimate of the sampling variance reduces the detection to 7.1σ and 8.1σ , respectively.We demonstrated that RCSLenS data improve the SNR of the measurements significantly compared toprevious studies where CFHTLenS data were used.The Fourier-space analysis, while requiring significant processing to account for masking effects, ismore useful for probing the impact of physical effects at different scales. We work with Cy−κ` and testfor consistency of the results with the configuration-space estimators. We reach similar conclusions forthe Cy−κ` measurement as the configuration-space counterpart as well as the same level of significanceof detection.The high level of detection compared to similar measurements in van Waerbeke et al. (2014) is dueto two main improvements: the larger sky coverage offered by RCSLenS survey has suppressed thestatistical uncertainties of the measured signal; and the final tSZ map provided by the Planck team isalso less noisy than that used in van Waerbeke et al. (2014).We have compared our measurements against predictions from the halo model, which adopts theempirically-motivated ‘universal pressure profile’ to describe the pressure of the hot gas associatedwith haloes. We highlighted the difficulties in estimating the covariance matrix for the type of cross-correlation measurement we consider in this chapter. Our error analysis includes contributions fromboth statistical uncertainty and sampling variance. The large sampling variance in the cross-correlationsignal originates mainly from the tSZ maps due to the dependence of the tSZ signal on the halo mass,AGN feedback and other stochastic processes. We estimated its contribution from field-to-field variance,but recognize that better estimation requires access to more data or more hydrodynamical simulations.Predictions from a WMAP-7yr best-fit cosmology match the data better than those based on the Planckcosmology, in agreement with previous studies (e.g., McCarthy et al. 2014).Finally, we employed the cosmo-OWLS hydrodynamical simulations (Le Brun et al., 2014), usingsynthetic tSZ and weak lensing maps produced for a wide range of baryonic physics models in boththe WMAP-7yr and Planck cosmologies. In agreement with the findings of the halo model results,72the comparison to the predictions of the simulations yields a preference for the WMAP-7yr cosmologyregardless of which feedback model is adopted. This is noteworthy, given the vast differences in themodels in terms of their predictions for the ICM properties of groups and clusters (which bracket theobserved hot gas properties of local groups and clusters, see Le Brun et al., 2014). The detailed shapeof the measured cross-correlations tend to prefer models that invoke significant feedback from AGN,consistent with what is found from the analysis of observed scaling relations, although there is stillsome degeneracy between the adopted cosmological parameters and the treatment of feedback physics.Future high-resolution CMB experiments combined with large sky area from a galaxy survey can inprinciple break the degeneracy between feedback models and place tighter constrains on the modelparameters.We highlighted the difficulties in estimating the covariance matrix for the type of cross-correlationmeasurement we consider in this chapter. The large sampling variance in the cross-correlation signal,which originates mainly from the tSZ maps, requires access to more data or more hydrodynamical sim-ulations to be accurately estimated. With the limited data we have, the covariance matrix that containedthe contribution from sampling variance was noisy, making it impossible to perform a robust signifi-cance analysis of the measurements. This is an area where further effort is required and we will pursuethis in future work.73Chapter 4Weak lensing corrections to tSZ-lensingcross-correlationThe cross-correlation between the thermal Sunyaev-Zeldovich (tSZ) effect and gravitational lensingin wide fields has recently been measured. This can be used to probe the distribution of the diffusegas in large-scale structure, as well as inform us about the missing baryons. As for any lensing-basedquantity, higher-order lensing effects can potentially affect the signal. Here, we extend previous higher-order lensing calculations to the case of tSZ-lensing cross-correlations. We derive terms analogous tocorrections due to the Born approximation, lens-lens coupling, and reduced shear up to order O(Φ4)in the Newtonian potential. Redshift distortions and vector modes are shown to be negligible at thisorder. We find that the dominant correction due to the reduced shear exceeds the percent-level only atmultipoles of `& 3000.4.1 IntroductionEven though both the direct detection of dark matter and its microscopic description have proven to beelusive so far, its macroscopic behaviour is thought to be well understood. The large-scale clustering ofdark matter has been observed though gravitational lensing and has been found to agree with theoreticalpredictions; see Bartelmann (2010) for a review and Hildebrandt et al. (2014) for an overview of therecent results with the Canada France Hawaii Telescope Lensing Survey (CFHTLenS). On small scalesits clustering behaviour has been modelled with N-body simulations to relatively high precision (e.g.,Harnois-De´raps et al., 2015a). Conversely, even though the microscopic behaviour of baryons is fullyunderstood, half of the Universe’s baryon content is in a hitherto unobserved state (Fukugita et al., 2004;Bregman, 2007); the ‘missing baryon problem’.A significant fraction of these missing baryons might reside in a warm, low density phase beyondgalactic halos (Anderson et al., 2010). By cross-correlating the thermal Sunyaev-Zeldovich (tSZ) effectmaps, which traces warm electrons, and mass maps derived from weak gravitational lensing data, thereis now observational support for the possibility that a significant fraction of the baryons indeed residesin such a phase (van Waerbeke et al., 2014; Ma et al., 2015; Hill et al., 2014).74Future surveys with large sky coverage (de Jong et al., 2013a; Sa´nchez et al., 2010) will producedata whose precision warrants a more sophisticated theoretical treatment than has been necessary so far.In this work we investigate the effect of higher-order lensing terms on the tSZ-lensing cross-correlation.There has been considerable effort to characterize higher-order contributions to correlations of lensingobservables (Schneider et al., 1998; Cooray et al., 2002; Hirata et al., 2003; Dodelson et al., 2005b,2006; Shapiro et al., 2006; Shapiro, 2009; Krause et al., 2010; Bernardeau et al., 2010; Vanderveld et al.,2011; Bernardeau et al., 2012; Andrianomena et al., 2014). Some of these higher-order effects, like therotation power spectrum, have been successfully observed in high-resolution ray-tracing simulations(Hilbert et al., 2009; Becker, 2013). Building on this corpus of previous work, we derive analogouscontributions to the tSZ-lensing cross-correlation.In section 4.2 we introduce the notation and recapitulate the first-order results. In section 4.3 weinvestigate the higher-order corrections. The terms related to the Born approximation, i.e., the evalu-ation of the integrals along the unperturbed photon path, and lens-lens coupling are derived in section4.3.1. The observed quantity in weak gravitational lensing is the reduced shear. Corrections due to thenon-linear relation between the shear and the convergence are derived in section 4.3.2. We also con-sider redshift distortions in section 4.3.3 and vector modes in section 4.3.4 and show that they do notcontribute at the order we are considering.4.2 First orderIn the Newtonian gauge the perturbed Robertson-Walker (RW) metric without anisotropic stresses canbe written asds2 = a(η)2[−(1+2Φ)dη2+(1−2Φ)(dχ2+ fK(χ)2dΩ2)] , (4.1)with fK(χ) the comoving angular diameter distance and χ the comoving radial distance. We will hence-forth work in units where c = 1. The potential Φ is assumed to be small, i.e., Φ 1. The first-ordersolution to the geodesic deviation equation at a comoving distance χ from the observer is then (Schnei-der et al., 1992; Schneider et al., 1998; Bartelmann, 2010)xi(~θ ,χ) = fK(χ)θ i−2∫ χ0dχ ′ fK(χ−χ ′)Φ,i(~x(~θ ,χ ′),χ ′) , (4.2)where ~θ represents the angle between the perturbed and fiducial ray at the observer. Vector quantitiesare denoted by lowercase Latin indices and partial derivatives with respect to comoving transverse coor-dinates, i.e., those perpendicular to the line-of-sight, are denoted by a comma. We make use of the sumconvention where repeated indices are summed over. Unless otherwise noted, this sum only includesthe two transverse directions. The Jacobi map is defined as the derivative of the deflection angle ~x(~θ ,χ)fK(χ)with respect to ~θ , i.e.,Ai j(~θ ,χ) =∂xi(~θ ,χ)fK(χ)∂θ j= δi j−2∫ χ0dχ ′fK(χ−χ ′) fK(χ ′)fK(χ)Φ,ik(~x(~θ ,χ ′),χ ′)Ak j(~θ ,χ ′) . (4.3)75It can be expressed in terms of the convergence κ , shear γ1, γ2, and rotation ω asAi j =(1−κ− γ1 −γ2−ω−γ2+ω 1−κ+ γ1)= δi j−ψi j . (4.4)Here we have introduced the distortion tensor ψi j as a measure of the deviation from the identity map.The convergence is then given by the trace of the distortion tensor:κ =12ψii . (4.5)Using (4.3) and (4.5), we find for the first-order convergenceκ(1)(~θ ,χS) =∫ χS0dχ ′K(χS ,χ′)Φ,ii( fK(χ ′)~θ ,χ ′) , (4.6)where we have defined the kernelK(χS ,χ′) =fK(χS −χ ′) fK(χ ′)fK(χS)Θ(χS −χ ′) . (4.7)Equation (4.6) describes the convergence due to a single source at a comoving distance χS = χ(zS) fromthe observer. The convergence of a population of sources with redshift distribution n(z)dz is found byaveraging over the sources with n(z) as the weighting factor. One then findsκ(1)(~θ) =∫ ∞0dχ(z)p(z)dzdχκ(1)(~θ ,χ(z)) =∫ ∞0dχW κ(χ)Φ,ii( fK(χ)~θ ,χ) , (4.8)with the kernel given byW κ(χ) =∫ ∞χdχ ′p(z)dzdχ ′K(χ ′,χ) . (4.9)The tSZ effect involves the inverse Compton scattering of CMB photons off relativistic electrons(Sunyaev et al., 1972). This introduces a frequency dependent temperature shift ∆T in the observedCMB temperature. The temperature shift at position ~θ on the sky and frequency ν can be parameterizedas∆TT0(~θ ,ν) = y(~θ)SSZ(ν) , (4.10)where the Compton y(~θ) parameter encodes the spatial and SSZ(ν) the spectral dependence. The Comp-ton y parameter is defined as the line-of-sight integral over the electron pressure. In this work we adaptthe constant bias model of van Waerbeke et al. (2014) to simplify the analysis. It has been shown inMa et al. (2015) that the constant bias model is consistent with a halo model approach, thus justifyingthe use of the simpler model. For a constant bias, the y parameter can be written as an integral over the76density contrast δ , i.e.,y(~θ) =∫ χH0dχW˜ y(χ)δ (~x(~θ ,χ),χ) , (4.11)where χH is the comoving distance to the surface of last scattering. We express the density contrast interms of the Newtonian potential through the Poisson equation asy(~θ) =∫ χH0dχW˜ y(χ)2a∆Φ(~x(~θ ,χ),χ)3H20ΩM=∫ χH0dχW y(χ)∆Φ(~x(~θ ,χ),χ) , (4.12)where we have absorbed the factors from the Poisson equation into the new kernel W y(χ).Ultimately, we are interested in the angular cross-power spectrum Cyκ` . In this work we assumethat the convergence is derived from shear measurements of galaxy surveys. The sky coverage of thesesurveys is still relatively small (although this will not be the case of future surveys such as Euclid andLSST) allowing the analysis to proceed in the flat-sky approximation. Using the definition of the 2DFourier transform (C.2), we can write the angular cross-power spectrum of yˆ(~`) and κˆ(~`) as〈yˆ(1)(~`1)κˆ(1)(~`2)〉= (2pi)2δ 2D(~`1+~`2)C(2)`1=∫ χH0dχdχ ′W y(χ)W κ(χ ′)|~`1|2fK(χ)2|~`2|2fK(χ ′)2〈φˆ(~`1,χ)φˆ(~`2,χ ′)〉 ,(4.13)where we have dropped the contributions of the derivatives along the line-of-sight in (4.12). Underthe Limber approximation (Limber, 1953; Kaiser, 1992) one assumes that the transverse modes aremuch large than the longitudinal modes, i.e., `fK(χ)  k3. This ceases to be true on large scales, whereextensions to the Limber approximation such as those described in LoVerde et al. (2008) or an exactfull-sky treatment have to be employed. However, for the scales of interest in this work the Limberapproximation is sufficient. In fact, it has been shown in Bernardeau et al. (2012) that the lowest-order Limber approximation is an excellent fit down to multipoles of `≈ 20. Expressing the two-pointfunction in (4.13) in terms of the power spectrum (C.5) we find for the y–κ cross-power spectrum〈yˆ(1)(~`1)κˆ(1)(~`2)〉= (2pi)2δ 2D(~`1+~`2)∫ χH0dχW y(χ)W κ(χ)|~`1|4fK(χ)4PΦ(|~`1|fK(χ),χ). (4.14)Note that upon replacing the kernel for the y parameter W y with W κ , one recovers the well knownexpression for the angular power spectrum of the convergence.4.3 CorrectionsTo consistently treat fourth-order corrections to the cross spectrum we need to include terms up tothird order in Φ of the Compton y parameter and convergence κ (Cooray et al., 2002). Indeed, it hasbeen shown in Krause et al. (2010) that divergences in second-second order cross terms cancel withcorresponding divergences in first-third order cross terms. It is thus important to find expressions for thetwo fields y and κ up to third order. A full sky treatment of lensing observables to even second order is77already a formidable task (Bernardeau et al., 2010, 2012; Vanderveld et al., 2011); a full sky derivationto third order would be beyond the scope of this work. Fortunately, the calculations can be simplifiedgreatly by restricting ourselves to small scales. We follow Dodelson et al. (2005b) to identify the termsthat contribute dominantly at small scales and those that can be neglected.Broadly speaking, on small scales terms with the most angular derivatives are expected to domi-nate. At second order this are: the well known Born approximation; lens-lens coupling; and reducedshear contributions (Bernardeau et al., 1997; Schneider et al., 1998; Cooray et al., 2002; Dodelson et al.,2005b, 2006; Shapiro, 2009; Krause et al., 2010). Third order terms derived from the aforementioned ef-fects have at least the same number of angular derivatives and are therefore expected to be the dominantthird-order contributions. We discuss these contributions in sections 4.3.1 and 4.3.2.Recent work by Andrianomena et al. (2014) found that contributions from peculiar velocities to theconvergence can be as large as the primary contribution from scalar modes in certain redshift ranges.Even though peculiar velocities formally affect the convergence at first order, they affect the shear onlyat second order (Bonvin, 2008). In the case where the convergence is derived from shear measurements,as we assume in this work, the effect of peculiar velocities enters only at second order. We investigatethe effect of peculiar velocities in section 4.3.3. Vector modes induced by second-order perturbationshave been shown to yield corrections of similar magnitude as traditional Born and lens-lens terms (An-drianomena et al., 2014). We show that vector modes do not contribute to the y–κ cross spectrum atfourth order in section 4.3.4.To distinguish the different corrections to the convergence we denote them by subscripts: κstd refersto corrections due to Born approximation and lens-lens coupling; κrs to corrections due to the reducedshear; and κz and κv to corrections due to redshift distortions and vector modes, respectively.4.3.1 Born approximation and lens-lens couplingFor the derivation of the Born and lens-lens coupling terms we roughly follow Krause et al. (2010),in that we expand the solution to the geodesic deviation equation (4.2) systematically in powers ofΦ. Alternatively, one could expand the terms in the distortion matrix (4.3), which makes the physicalmeaning of the terms more apparent.We expand the comoving transverse displacement in powers of the potential Φ as~x =~x(0)+~x(1)+~x(2)+~x(3)+O(Φ4) , (4.15)where the superscript in parentheses denotes the order of the expansion. The zeroth and first-ordercontributions are given by~x(0) = fK(χ)~θ , x(1)i(~θ ,χ) =−2∫ χ0dχ ′ fK(χ−χ ′)Φ,i( fK(χ ′)~θ ,χ ′) . (4.16)The higher-order contributions can be found by Taylor expanding Φ(~x) in (4.2) around the zeroth ordersolution~x(0)(~θ ,χ) = fK(χ)~θ . The potential can then be expanded as Φ=Φ(1)+Φ(2)+Φ(3)+O(Φ4),78withΦ(1)(~x) =Φ(~x(0))Φ(2)(~x) =Φ,i(~x(0))x(1)iΦ(3)(~x) =12Φ,i j(~x(0))x(1)ix(1) j +Φ,i(~x(0))x(2)i .(4.17)By replacing the Φ with Φ(2) in (4.2), we can write the second-order deflection angle asx(2)i(~θ ,χ)fK(χ)=−2∫ χ0dχ ′fK(χ−χ ′)fK(χ)Φ(2),i (~x(~θ ,χ′),χ ′)= 4∫ χ0dχ ′∫ χ ′0dχ ′′K(χ,χ ′)K(χ ′,χ ′′)fK(χ ′′)Φ,i j(χ ′)Φ, j(χ ′′) ,(4.18)where we have dropped the angular dependence of the potentials for brevity. We adapt this shorthandfor the rest of this work, i.e., unless otherwise noted Φ(~x(0)(~θ ,χ),χ) is written as Φ(χ). Analogously,the third-order deflection angle can be written asx(3)i(~θ ,χ)fK(χ)=−4∫ χ0dχ ′∫ χ ′0dχ ′′∫ χ ′0dχ ′′′K(χ,χ ′)K(χ ′,χ ′′)K(χ ′,χ ′′′) fK(χ ′)fK(χ ′′) fK(χ ′′′)×Φ,i jk(χ ′)Φ, j(χ ′′)Φ,k(χ ′′′)−8∫ χ0dχ ′∫ χ ′0dχ ′′∫ χ ′′0dχ ′′′K(χ,χ ′)K(χ ′,χ ′′)K(χ ′′,χ ′′′)fK(χ ′′′)×Φ,i j(χ ′)Φ, jk(χ ′′)Φ,k(χ ′′′) .(4.19)ConvergenceEquipped with second- and third-order expressions for the deflection angle it is straightforward to deriveexpressions for the convergence. Using the relation of the convergence to the trace of the distortiontensor (4.5), we can readily write down the second- and third-order expressions for the convergence. Atsecond order this isκ(2)std (~θ ,χS) =−2∫ χS0dχ ′∫ χ ′0dχ ′′K(χ,χ ′)K(χ ′,χ ′′)×(fK(χ ′)fK(χ ′′)Φ,ii j(χ ′)Φ, j(χ ′′)+Φ,i j(χ ′)Φ, ji(χ ′′)).(4.20)The first term in the bracket is the well known Born term, while the second is the lens-lens couplingterm. The extra factors of the comoving angular distance fK(χ) arise because the derivative in (4.3) iswith respect to the angular deviation ~θ , whereas the potential is a function of the comoving transversedistance~x(0) = fK(χ)~θ .79The third-order expression for the convergence is analogously found to beκ(3)std (~θ ,χS) = 2∫ χS0dχ ′∫ χ ′0dχ ′′∫ χ ′0dχ ′′′K(χ,χ ′)K(χ ′,χ ′′)K(χ ′,χ ′′′)×(fK(χ ′)2fK(χ ′′) fK(χ ′′′)Φ,ii jk(χ ′)Φ, j(χ ′′)Φ,k(χ ′′′))+4∫ χS0dχ ′∫ χ ′0dχ ′′∫ χ ′0dχ ′′′K(χ,χ ′)K(χ ′,χ ′′)K(χ ′,χ ′′′)×(fK(χ ′)fK(χ ′′′)Φ,i jk(χ ′)Φ, ji(χ ′′)Φ,k(χ ′′′))+4∫ χS0dχ ′∫ χ ′0dχ ′′∫ χ ′′0dχ ′′′K(χ,χ ′)K(χ ′,χ ′′)K(χ ′′,χ ′′′)fK(χ ′′′)× ∂∂θ i(Φ,i j(χ ′)Φ, jk(χ ′′)Φ,k(χ ′′′)).(4.21)The term on line 2 corresponds a second-order Born correction, the term on line 4 to a mixed Born-lens-coupling, and the three terms on line 6 to a second-order Born correction, Born-lens-coupling,second-order lens-lens coupling, respectively.Compton y parameterThe second and third-order contributions to the Compton y parameter are somewhat easier to derive, asthere are no lens-lens coupling terms. As in the case of the convergence, we replace Φ in (4.12) by itsexpansion (4.17). The second-order contribution to the y parameter is theny(2)(~θ) =−2∫ χH0dχ ′∫ χ ′0dχ ′′W y(χ ′)K(χ ′,χ ′′)fK(χ ′)fK(χ ′′)Φ,ii j(χ ′)Φ, j(χ ′′) . (4.22)The third-order term follows analogously and is given byy(3)(~θ) = 2∫ χH0dχ ′∫ χ ′0dχ ′′∫ χ ′0dχ ′′′W y(χ ′)K(χ ′,χ ′′)K(χ ′,χ ′′′)fK(χ ′)2fK(χ ′′) fK(χ ′′′)×Φ,ii jk(χ ′)Φ, j(χ ′′)Φ,k(χ ′′′)+4∫ χH0dχ ′∫ χ ′0dχ ′′∫ χ ′′0dχ ′′′W y(χ ′)K(χ ′,χ ′′)K(χ ′′,χ ′′′)fK(χ ′)fK(χ ′′′)×Φ,ii j(χ ′)Φ, jk(χ ′′)Φ,k(χ ′′′) .(4.23)Both terms are due to the Born approximation. The term on line 2 stems from the 12Φ,i j(~x(0))x(1)ix(1) jterm in the third-order contribution to Φ in (4.17), whereas the term on line 4 in (4.23) is due to theΦ,i(~x(0))x(2)i term in (4.17).80Cross correlationsThe second-second-order contribution to the angular y–κ cross-power spectrum due to Born and lens-lens terms can be derived by taking the ensemble average of the product of the Fourier space expressionsyˆ(2)(~`1) and κˆ(2)(~`2). Using the results from appendix C.1, we find〈yˆ(2)(~`1)κˆ(2)std (~`2)〉= 4∫ χH0dχydχκ∫ χy0dχ ′y∫ χκ0dχ ′κW y(χy)W κ(χκ)K(χy,χ ′y)K(χκ ,χ ′κ)fK(χy)2 fK(χ ′y)2 fK(χκ)2 fK(χ ′κ)2×∫ d2~`′d2~`′′(2pi)4|~`′|2~`′(~`1−~`′)[|~`′′|2~`′′(~`2− ~`′′)+(~`′′(~`2− ~`′′))2]×〈φˆ(~`′,χy)φˆ(~`1−~`′,χ ′y)φˆ(~`′′,χκ)φˆ(~`2− ~`′′,χ ′κ)〉 ,(4.24)where we used the kernel W κ for a source distribution n(z) instead of a single source at redshift zS.The four-point function on the last line is made up of one connected and three unconnected terms. Theconnected term is proportional to the trispectrum (C.7). Under the Limber approximation this introducesa product of delta functions δD(χy− χ ′y)δD(χy− χκ)δD(χy− χ ′κ), setting all comoving distances alongthe line-of-sight equal. The kernel K(χ,χ ′) is zero for χ ≤ χ ′, thus eliminating the contribution from theconnected part of the correlation function. The unconnected part can be decomposed into three productsof two-point functions by Wick’s theorem. Each of the two-point functions yields a delta function timesa power spectrum. The term proportional to δD(χy−χ ′y)δD(χκ −χ ′κ) is zero because K(χ,χ) = 0. Theterm proportional to δD(χy−χ ′κ)δD(χ ′y−χκ) is zero because K(χ,χ ′)K(χ ′,χ)≡ 0. The only survivingterm is proportional to δD(χy−χκ)δD(χ ′y−χ ′κ), and upon evaluating the integrals gives for the angularcross-power spectrum〈yˆ(2)(~`1)κˆ(2)std (~`2)〉= 4(2pi)2δ 2D(~`1+~`2)∫ χH0dχ∫ χ0dχ ′W y(χ)W κ(χ)K(χ,χ ′)2fK(χ)6 fK(χ ′)6∫ d2~`′(2pi)2×|~`′|2~`1~`′(~`′(~`1−~`′))2PΦ(|~`′|fK(χ),χ)PΦ(|~`1−~`′|fK(χ ′),χ ′).(4.25)The derivation for the first-third order contributions proceeds similarly. The connected correlation func-tion drops out for the same reason as in the second-second-order case. Furthermore, terms in the third-order expressions for y and κ that include a line-of-sight kernel proportional to K(χ ′,χ ′′)K(χ ′′,χ ′′′),i.e., line 5 in (4.21) and line 3 in (4.23), do not contribute to the power spectra because the kernel is zerofor all possible contractions of the correlation function.The contribution from the second term in (4.21), i.e., line 4, to the cross-power spectrum is propor-tional to ∫ d2~`′(2pi)2(~`1~`′)3PΦ(|~`1|fK(χ),χ)PΦ(|~`′|fK(χ ′),χ ′),which is zero due to the antisymmetry of the integral under the transformation ~`′→−~`′ (Krause et al.,812010). Hence, only the first Born term in (4.21) contributes to 〈yˆ(1)κˆ(3)〉. We find〈yˆ(1)(~`1)κˆ(3)std (~`2)〉=−2(2pi)2δ 2D(~`1+~`2)∫ χH0dχ∫ χ0dχ ′W y(χ)W κ(χ)K(χ,χ ′)2fK(χ)6 fK(χ ′)6×∫ d2~`′(2pi)2|~`1|4(~`1~`′)2PΦ(|~`1|fK(χ),χ)PΦ(|~`′|fK(χ ′),χ ′).(4.26)Since the only contribution to 〈yˆ(3)κˆ(1)〉 comes from the first term in (4.23), which is identical to the firstterm in (4.21) up to an interchange of the kernels W y(χ) and W κ(χ), the cross-power spectra 〈yˆ(3)κˆ(1)〉and 〈yˆ(1)κˆ(3)〉 are identical.4.3.2 Reduced shearAt first order the shear and convergence are related byκˆ(~`) = T I(~`)γˆI(~`) , T 1(~`) = cos2φ` , T 2(~`) = sin2φ` , (4.27)where φ` is the angle between the two-dimensional wave-vector ~` and some fixed reference axis. Thecomponents of the shear and other polar quantities are labeled by capital Latin indices. It can be shownthat this relation holds exactly up to second order and under the Limber approximation up to third order(see appendix C.2 for details). In the weak lensing regime, the measured quantity is not the shear itselfbut the reduced shear, conventionally defined asgI =γI1−κ , I = 1,2 . (4.28)This definition of the reduced shear is based on the assumption that the Jacobi map (4.3) is symmet-ric. In general the Jacobi map is not symmetric however, because lens-lens couplings generate theanti-symmetric contribution ω . Including the anti-symmetric terms in the Jacobi map, the generalizedreduced shear in complex notation is given by (see appendix C.3)g =γ1+ iγ21−κ+ iω . (4.29)Accounting for the reduced shear in the relation (4.27) amounts to replacing the shear γI with the reducedshear gI . Using (C.22) and expanding systematically in Φ to third order we can express the observedconvergence asκˆobs = T I gˆI = T I(γˆ(1)I +(γˆ(2)std )I +(γˆ(3)std )I+ γˆ(1)I ∗ κˆ(1)+ γˆ(1)I ∗ κˆ(1) ∗ κˆ(1)+ γˆ(1)I ∗ κˆ(2)std +(γˆ(2)std )I ∗ κˆ(1)+R(ωˆ(2)std )IJ ∗ γˆ(1)J)+O(Φ4) ,(4.30)82where ∗ stands for a convolution in Fourier space. As shown in appendix C.2, the first line is equivalentto κˆ(1)+ κˆ(2)std + κˆ(3)std , where κˆ(2)std and κˆ(3)std denote the corrections due to Born approximation and lens-lens coupling. The second line includes the well known contributions from the reduced shear (Schneideret al., 1998; Dodelson et al., 2006; Shapiro, 2009; Krause et al., 2010), while the third line is a novelcontribution due to second-order induced rotations. The sole second-order correction due to reducedshear to the convergence isκˆ(2)rs (~`) = [T I(~`)γˆ(1)I ∗ κˆ(1)](~`) =∫ d2~`′(2pi)2T I(~`)γˆ(1)I (~`′)κˆ(1)(~`−~`′)=∫ d2~`′(2pi)2cos(2φ`′−2φ`)κˆ(1)(~`′)κˆ(1)(~`−~`′) ,(4.31)where we used the identity T I(~`)TI(~`′) = cos(2φ`′−2φ`). Since the reduced shear is an intrinsic lensingeffect, it does not affect the Compton y parameter. The lowest-order contribution to the cross-powerspectrum is therefore formed by the first-order y parameter (4.12) and second-order reduced shear cor-rection (4.31), i.e.,〈yˆ(1)(~`1)κˆ(2)rs (~`2)〉=−(2pi)2δ 2D(~`1+~`2)∫dχW y(χ)(W κ(χ))2fK(χ)10∫ d2~`′(2pi)2×|~`1|2|~`′|2|~`2−~`′|2 cos(2φ`′−2φ`2)BΦ(|~`1|fK(χ),|~`′|fK(χ),|~`2−~`′|fK(χ)),(4.32)where we used the definition (C.6) of the bispectrum. Unlike in the case of the Born and lens-lens terms,there is a third-order contribution to the cross-power spectrum. As a consistency check, one can showthat upon replacing W y by W κ , and using the fact that to first order the convergence is the same as theE-mode of the shear, one recovers the expression for the correction to the E-mode shear due to reducedshear in Dodelson et al. (2006).To analyze the first-third-order contributions, we split the third-order contribution to the convergencedue to the reduced shear in (4.30) into three components:κˆ(3,A)rs (~`) = T I(~`)[γˆ(1)I ∗ κˆ(1) ∗ κˆ(1)](~`) ; (4.33a)κˆ(3,B)rs (~`) = T I(~`)[γˆ(1)I ∗ κˆ(2)std +(γˆ(2)std )I ∗ κˆ(1)](~`) ; (4.33b)κˆ(3,C)rs (~`) = T I(~`)[R(ωˆ(2)std )IJ ∗ γˆ(1)J ](~`) . (4.33c)The cross-power spectrum of κˆ(3,A)rs with y is then〈yˆ(1)(~`1)κˆ(3,A)rs (~`2)〉=∫dχyW y(χy)fK(χy)23∏i=1dχiW κ(χi)fK(χi)2∫ d2~`′d2~`′′(2pi)4cos(2φ`2−2φ`′)×|~`1|2|~`′|2|~`′′|2|~`2−~`′− ~`′′|2×〈φˆ(~`1,χy)φˆ(~`′,χ1)φˆ(~`′′,χ2)φˆ(~`2−~`′− ~`′′,χ3)〉 .(4.34)83Because the line-of-sight integral does not include the kernel K(χ,χ ′), like in the case of the third-order cross-power spectrum (4.32), the connected part of the four-point function does not vanish. Theconnected and unconnected contributions to the cross-power spectrum are found to be〈yˆ(1)(~`1)κˆ(3,A)rs (~`2)〉c = (2pi)2δ 2D(~`1+~`2)∫dχW y(χ)W κ(χ)3fK(χ)14∫ d2~`′d2~`′′(2pi)4cos(2φ`2−2φ`′)×|~`1|2|~`′|2|~`′′|2|~`1+~`′+ ~`′′|2TΦ(~`1fK(χ),~`′fK(χ),~`′′fK(χ),−~`1+~`′+ ~`′′fK(χ),χ)(4.35a)〈yˆ(1)(~`1)κˆ(3,A)rs (~`2)〉g = 〈yˆ(1)(~`1)κˆ(1)(~`2)〉∫ d2~`′(2pi)2Cκκ,2`′ , (4.35b)where the connected and unconnected parts are denoted by the subscript c and g, respectively. For thederivation of the connected part we have used the definition of the trispectrum C.7. Replacing the kernelW y by W κ we recover again the same expression as found in Krause et al. (2010).The two other third-order contributions to the convergence due the reduced shear (4.33b) and (4.33c)involve second-order Born and lens-lens corrections, i.e., include the coupling kernel K(χ,χ ′) in theirline-of-sight integrals. Hence, only their unconnected parts contribute to the cross-power spectrum. Thederivation proceeds as for the other terms discussed so far, albeit with somewhat more complicatedexpressions, as there are now two contractions of the four-point function that survive. The contributioninvolving κˆ(3,B)rs is〈yˆ(1)(~`1)κˆ(3,B)rs (~`2)〉=−2(2pi)2δ 2D(~`1+~`2)∫dχdχ ′W y(χ)W κ(χ ′)fK(χ)6 fK(χ ′)6∫ d2~`′(2pi)2×|~`1|2|~`′|2~`1~`′PΦ(|~`1|fK(χ),χ)PΦ(|~`′|fK(χ ′),χ ′)×{cos(2φ`2−2φ`′)~`1~`′[W κ(χ)K(χ,χ ′)+W κ(χ ′)K(χ ′,χ)]cos(2φ`2−2φ~`′+~`1)[~`1(~`′+~`1)W κ(χ)K(χ,χ ′)+~`′(~`′+~`1)W κ(χ ′)K(χ ′,χ)]}.(4.36)The azimuthal integral of the third line can be done analytically and is equal to pi2 . Note that our resultdiffers from that obtained in Krause et al. (2010) by an extra factor of cos(2φ`2 −2φ~`′+~`1) on the fourthline.Using the definition of the matrix R(ω) in (C.23), the contribution κˆ(3,C)rs can be written asκˆ(3,C)rs (~`) = T 1(~`)[γ(1)2 ∗ ωˆ(2)std ](~`)−T 2(~`)[γ(1)1 ∗ ωˆ(2)std ](~`)=∫ d2~`′(2pi)2sin(2φ`′−2φ`)κˆ(1)(~`′)ωˆ(2)std (~`−~`′) .(4.37)84The cross-power spectrum is therefore〈yˆ(1)(~`1)κˆ(3,C)rs (~`2)〉=−2(2pi)2δ 2D(~`1+~`2)∫dχdχ ′W y(χ)W κ(χ ′)fK(χ)6 fK(χ ′)6∫ d2~`′(2pi)2×|~`1|3|~`′|3~`1~`′ sin(φ`2−φ`′)sin(2φ`′−2φ`2)PΦ(|~`1|fK(χ),χ)PΦ(|~`′|fK(χ ′),χ ′)× [W κ(χ)K(χ,χ ′)+W κ(χ ′)K(χ ′,χ)] .(4.38)The mode-coupling term can be reduced to |~`1|4|~`′|4 sin(2φ`′−2φ`2 )22 . The azimuthal integral then evaluatesto pi2 . The contributions from κˆ(3,C)rs and from the first term in κˆ(3,B)rs are therefore identical.Finally, we find the only second-second-order contribution due to the reduced shear to the cross-power spectrum to be〈yˆ(2)(~`1)κˆ(2)rs (~`2)〉=−2(2pi)2δ 2D(~`1+~`2)∫dχdχ ′W y(χ)K(χ,χ ′)W κ(χ)W κ(χ ′)fK(χ)6 fK(χ ′)6∫ d2~`′(2pi)2×|~`′|4|~`1−~`′|2[~`′(~`1−~`′)]PΦ(|~`′|fK(χ),χ)PΦ(|~`1−~`′|fK(χ ′),χ ′)×[cos(2φ`2−2φ~`′)+ cos(2φ`2−2φ~`1−~`′)].(4.39)The dominant contribution is the third-order correction (4.32), by virtue of being of a lower order thanthe other contributions considered in this work, which are all of fourth order.4.3.3 Redshift distortionsThe comoving line-of-sight distance to a source is usually not an observable quantity. Instead it isderived from the measured redshift, which is affected by the peculiar motions of the source and observer,Sachs-Wolfe, and integrated Sachs-Wolfe effects. The second-order contribution to the convergence dueto a perturbation of the cosmological redshift isκ(2)z (χ) =dκ(1)(χ)dzδ z(1) =dκ(1)(χ)dχdχdzδ z(1) . (4.40)The dependence of the convergence on comoving distance of the source isdκ(1)(χ)dχ=1fK(χ)2∫ χ0dχ ′Φ,ii(χ ′)fK(χ ′)2, (4.41)while the redshift perturbation due to peculiar motion of the source, Sachs-Wolfe, and integrated Sachs-Wolfe effects is given by (Bernardeau et al., 2010)δ z(1) =1a(−2∫ χ0dχ ′∂Φ(χ ′)∂χ ′+Φ(χ)−niv(1)i (χ)), (4.42)85where the potential at the observer and the peculiar motion of the observer have been set to zero, asthey would only contribute at the very largest scales. The peculiar motion from first-order perturbationtheory is (Bernardeau et al., 2012)v(1)i (χ) =−2a3H20Ωm∂i(−∂Φ(χ′)∂χ ′+H (χ)Φ(χ)). (4.43)From (4.42) and (4.43) we can already see that only the term corresponding to the peculiar motion wouldcontribute appreciably, as it involves an angular derivative. Restricting ourselves to the contribution dueto the peculiar motion, the second-order convergence can be written asκ(2)z (χ) =−niv(1)i (χ)H (χ) fK(χ)2∫ χ0dχ ′Φ,ii(χ ′)fK(χ ′)2. (4.44)The photon trajectory~n projects the peculiar velocity along the line-of-sight, i.e., the angular derivativesare projected out. Thus all redshift distortions that contribute to (4.40) have only two angular derivativesand can be safely neglected on small scales.4.3.4 Vector modesIn Andrianomena et al. (2014) it was shown that fourth-order contributions from vector modes to lensingobservables can be of comparable magnitude as other fourth-order contributions considered in this work.It would thus be conceivable that there are large third-order contributions involving vector modes. Thelowest-order cross-correlation that includes vector modes is 〈yˆ(1)(~`1)κˆ(2)v (~`2)〉, where the second-ordercontribution to the convergence is (Andrianomena et al., 2014; Thomas et al., 2015)κ(2)v (χ) =∫ χ0dχ ′K(χ,χ ′)n jV j,ii(χ′) . (4.45)The contraction of the line-of-sight direction~n with the vector potential V i is proportional toniV i(χ) ∝ sinϑ e±iϕ , (4.46)where ϑ and ϕ denote the spherical coordinates on the sky. This expression is manifestly of odd parityand does not contribute if one correlates it with the even parity field y. The lowest-order vector contri-bution to the cross-power spectrum has to be quadratic in the vector potential. Since the vector potentialis already of second order in the scalar potential Φ, and the lowest vector contribution to y is of thirdorder, there are no fourth-order vector contribution to the cross-power spectrum.4.4 DiscussionIn Fig. 4.1 we have plotted the cross-power spectrum (4.14) and the various higher-order contributionsconsidered in this work. The underlying non-linear matter power spectrum was computed with CAMB,11http://camb.info86102 103 104`10−1310−1210−1110−1010−910−810−710−6`(`+1)Cyκ`/2piCyκ`Cosmic varianceReduced shear O(Φ3)Reduced shear O(Φ4)Born apprioximation& lens-lens couplingFigure 4.1: The different contributions to the angular cross-power spectrum Cyκ` . The first-orderresult (4.14)(bold blue), third-order reduced shear (4.32)(red), fourth-order Born and lens-lens terms (4.25) and (4.26)(cyan), and fourth-order reduced shear contributions (4.35),(4.36), (4.38), (4.39) (green).using the best fit Planck cosmological parameters (Planck Collaboration XVI, 2014) and the extension toHALOFIT by Takahashi et al. (2012). For the source redshift distribution n(z) we use the fitting formulaof the redshift distribution of CFHTLenS (van Waerbeke et al., 2013). We computed the non-linearbispectrum using the fitting formulae of both Scoccimarro et al. (2001) and Gil-Marı´n et al. (2012). Itwas found in Fu et al. (2014) that the fitting formula in Gil-Marı´n et al. (2012) slightly overestimatesthe bispectrum on small scales compared to that of Scoccimarro et al. (2001). For clarity, we only showthe reduced shear contribution computed with the fitting formula of Scoccimarro et al. (2001) in Fig.4.1. The relative contributions to the cross-power spectrum due to the third-order term (4.32) with bothfitting formulae is shown in Fig. 4.2. We find that the third-order contribution (4.32) gives the largestcorrection to the cross-power spectrum. At multipoles of `≈ 4000 it begins to dominate over the cosmicvariance((Cyκ` )2+Cyy` Cκκ`)/(2`+ 1) (Kamionkowski et al., 1997) and reaches several percent of thesecond-order result (4.14) at multipoles of `≈ 104. The fourth-order contributions are over an order ofmagnitude lower at small scales. Furthermore, the difference between the two fitting formulae for thebispectrum of Scoccimarro et al. (2001) and Gil-Marı´n et al. (2012) are at least an order of magnitudelarger on small scales than the fourth-order corrections. It is thus justified to approximate the fourth-order contribution (4.34) by its unconnected part, as it is expected to dominate over the connected partat all but the smallest scales (Krause et al., 2010).4.5 ConclusionWe have calculated all contributions up to fourth order due to weak lensing to the tSZ-lensing cross-correlation in the small angle approximation. We have found that only the third-order term (4.32) due tothe reduced shear contributes appreciably. At multipoles of `≈ 3000 the contribution reaches the percent87102 103 104`δCyκ`/Cyκ`Reduced shear SC01Reduced shear GM12Cosmic varianceFigure 4.2: The third-order contribution (4.32) to the cross-power spectrum computed using the fit-ting formulae for the bispectrum from Scoccimarro et al. (2001) and Gil-Marı´n et al. (2012).The corrections begin to dominate over cosmic variance above `∼ 4000.level and rises strongly from there. The effect might thus be observable in future high-resolution surveys,in particular for cluster samples where the tSZ-lensing cross-correlation signal will be measured aroundclusters and stacked. However, baryon physics can have similar — or even stronger — effects on thetSZ-lensing cross-correlation at small scales, such that the contributions calculated in this chapter mightremain unobservable for the foreseeable future. For upcoming large-area surveys such as LSST2 andEuclid3, a full-sky treatment will be necessary. As is evident from the large amount of terms in eventhe second-order shear in Bernardeau et al. (2010, 2012), a derivation to the same order as consideredin this work will be a formidable task.Even though the simple bias model employed in this work is compatible with a halo model approach(Ma et al., 2015), a treatment of the corrections considered in this work in the context of the halo modelwould be of interest. It should be noted that even within the framework of the halo model there is stillconsiderable uncertainty in the modelling of the pressure profile, exemplifying the complications oneencounters once baryonic physics are introduced.Beyond its explicit application to the cross-correlation between lensing and tSZ, our work can beused to calculate high order lensing corrections to cross-correlation signals other than tSZ. For instanceone can envision measuring the cross-correlation between the cosmic infrared background and lowerredshift structures, which in principles could require similar corrections to the cross-correlation withtSZ presented in this chapter.2http://www.lsst.org3http://sci.esa.int/euclid/88Chapter 5KiDS-450: tomographic cross-correlationof galaxy shear with Planck lensingWe present the tomographic cross-correlation between galaxy lensing measured in the Kilo-Degree Sur-vey (KiDS-450) with overlapping lensing measurements of the cosmic microwave background (CMB),as detected by Planck (2015 data release). We compare our joint probe measurement to the theoreticalexpectation for a flat ΛCDM cosmology, assuming the best-fitting cosmological parameters from theKiDS-450 cosmic shear and Planck CMB analyses. We find that our results are consistent within 1σwith the KiDS-450 cosmology, with an amplitude re-scaling parameter AKiDS = 0.86±0.19. Adoptinga Planck cosmology, we find our results are consistent within 2σ , with APlanck = 0.68±0.15. We showthat the agreement is improved in both cases when the contamination of the signal by intrinsic galaxyalignments is accounted for, increasing A by approximately 0.1. This is the first tomographic analysisof the galaxy lensing – CMB lensing cross-correlation signal, and is based on five photometric redshiftbins. We use this measurement as an independent validation of the multiplicative shear calibration andof the calibrated source redshift distribution at high redshifts. We find that constraints on these twoquantities are strongly correlated when obtained from this technique, which should therefore not beconsidered as a stand-alone competitive calibration tool.5.1 IntroductionRecent observations of distinct cosmological probes are closing in on the few parameters that enter thestandard model of cosmology (see for example Planck Collaboration XIII, 2016, and references therein).Although there is clear evidence that the Universe is well described by the ΛCDM model, some tensionsare found between probes. For instance, the best fit cosmology inferred from the observation of theCosmic Microwave Background (CMB) in Planck Collaboration XIII (2016) is in tension with somecosmic shear analyses (MacCrann et al., 2015; Hildebrandt et al., 2017; Joudaki et al., 2016, 2017),while both direct and strong lensing measurements of today’s Hubble parameter H0 are more than 3σaway from the values inferred from the CMB (Bernal et al., 2016; Bonvin et al., 2017). At face value,these discrepancies either point towards new physics (for a recent example, see Joudaki et al., 2016) or89un-modelled systematics in any of those probes. In this context, cross-correlation of different cosmicprobes stands out as a unique tool, as many residual systematics that could contaminate one data set areunlikely to correlate also with the other (e.g. ‘additive biases’). This type of measurement can thereforebe exempt from un-modelled biases that might otherwise source the tension. Another point of interest isthat the systematic effects that do not fully cancel, for example ‘multiplicative biases’ or the uncertaintyon the photometric redshifts, will often impact differently the cosmological parameters compared to thestand-alone probe, allowing for degeneracy breaking or improved calibration.In this chapter, we present the first tomographic cross-correlation measurement between CMB lens-ing and galaxy lensing, based on the lensing map described in Planck Collaboration XV (2016) andthe lensing data from the Kilo-Degree Survey1 presented in Kuijken et al. (2015, KiDS hereafter) andin the KiDS-450 cosmic shear analysis (Hildebrandt et al., 2017). The main advantage in this sort ofmeasurement resides in it being free of uncertainty on galaxy bias, which otherwise dominates the errorbudget in CMB lensing - galaxy position cross-correlations (Omori et al., 2015; Giannantonio et al.,2016; Baxter et al., 2016). Over the last two years, the first lensing-lensing cross-correlations were usedto measure σ8 and Ωm (Hand et al., 2015; Liu et al., 2015), by combining the CMB lensing data fromthe Atacama Cosmology Telescope (Das et al., 2014) with the lensing data from the Canada-France-Hawaii Telescope Stripe 82 Survey (Moraes et al., 2014), and from the Planck lensing data and theCanada-France-Hawaii Telescope Lensing Survey (Erben et al., 2013, CFHTLenS hereafter). Sincethen, additional effects were found to contribute to the measurement, introducing extra complicationsin the interpretation of the signal. For instance, Hall et al. (2014) and Troxel et al. (2014) showed thatthe measurement is likely to be contaminated by the intrinsic alignment of galaxies with the tidal fieldin which they live. At the same time, Liu et al. (2016) argued that this measurement could point insteadto residual systematics in the multiplicative shear bias and proposed that the measurement itself couldbe used to set constraints on the shear bias (see also Das et al., 2013). Their results showed that largeresiduals are favoured, despite the calibration accuracy claimed by the analysis of image simulationstailored for the same survey (Miller et al., 2013). A recent analysis from Harnois-De´raps et al. (2016,HD16 hereafter) suggested instead that the impact of catastrophic redshift outliers could be causing thisapparent discrepancy, since these dominate the uncertainty in the modelling. They also showed thatchoices concerning the treatment of the masks can lead to biases in the measured signal, and that thecurrent estimators should therefore be thoroughly calibrated on full light-cone mocks.Although these pioneering works were based on Fourier space cross-correlation techniques, morerecent analyses presented results from configuration-space measurements, which are cleaner due to theirinsensitivity to masking. Kirk et al. (2016, K16 hereafter) combined the CMB lensing maps from Planckand from the South Pole Telescope (van Engelen et al., 2012, SPT) with the Science Verification Datafrom the Dark Energy Survey2. Their measurement employed the POLSPICE numerical tool (Szapudiet al., 2001a; Chon et al., 2004), which starts off with a pseudo-C` measurement that is converted intoconfiguration space to deal with masks, then is turned back into a Fourier space estimator. Soon after,1KiDS: http://kids.strw.leidenuniv.nl2DES: www.darkenergysurvey.org90HD16 showed consistency between pseudo-C` analyses and configuration space analyses of two-pointcorrelation functions, combining the Planck lensing maps with both CFHTLenS and the Red-sequenceCluster Lensing Survey (Hildebrandt et al., 2016, RCSLenS hereafter). A similar configuration spaceestimator was recently used with Planck lensing and SDSS shear data (Singh et al., 2017), although thesignal was subject to higher noise levels.This chapter directly builds on the K16 and HD16 analyses, utilising tools and methods describedtherein, but on a new suite of lensing data. The additional novelty here is that we perform the firsttomographic cross-correlation analysis between CMB lensing and galaxy lensing, where we split thegalaxy sample into five redshift bins and examine the redshift evolution. This is made possible bythe high quality of the KiDS photometric redshift data, by the extent of the spectroscopic matchedsample, and consequently by the precision achieved on the calibrated source redshift distribution (seeHildebrandt et al., 2017, for more details). It provides a new test of cosmology within the ΛCDMmodel, including the redshift evolution of the growth of structure, and also offers an opportunity toexamine the tension between the KiDS and Planck cosmologies (reported in Hildebrandt et al., 2017).With the upcoming lensing surveys such as LSST3 and Euclid4, it is expected that this type of cross-correlation analysis will be increasingly used to validate the data calibration (Schaan et al., 2017) andextract cosmological information in a manner that complements the cosmic shear and clustering data.The basic theoretical background upon which we base our work is laid out in Section 5.2. Wethen describe the data sets and our measurement strategies in Sections 5.3 and 5.4, respectively. Ourcosmological results are presented in Section 5.5. We also describe therein a calibration analysis alongthe lines of Liu et al. (2016), this time focussing on high redshift galaxies for which the photometricredshifts and shear calibration are not well measured. Informed on cosmology from lower redshiftmeasurement, this self-calibration technique has the potential to constraint jointly the shear bias and thephoto-z distribution, where other methods fail. We conclude in Section 5.6.The fiducial cosmology that we adopt in our analysis corresponds to the flat WMAP9+SN+BAOcosmology5 (Hinshaw et al., 2013), in which the matter density, the dark energy density, the baryonicdensity, the amplitude of matter fluctuations, the Hubble parameter and the tilt of the matter power spec-trum are described by (Ωm,ΩΛ,Ωb,σ8,h,ns) = (0.2905,0.7095,0.0473,0.831,0.6898,0.969). Asidefrom determining the overall amplitude of the theoretical signal from the (σ8,Ωm) pair, this choice haslittle impact on our analysis, as we later demonstrate. Future surveys will have the statistical powerto constrain the complete cosmological set, but this is currently out of reach for a survey the size ofKiDS-450. We note that our fiducial cosmology is a convenient choice that is consistent within 2σ withthe Planck, KiDS-450, CFHTLenS, and WMAP9+ACT+SPT analyses in the (σ8,Ωm) plane. As such,it minimizes the impact of residual tension across data sets.3www.lsst.org4sci.esa.int/euclid5Our fiducial cosmology consists of a flat ΛCDM universe in which the dark energy equation of state is set to w =−1.915.2 Theoretical backgroundPhotons from the surface of last scattering are gravitationally lensed by large-scale structures in theUniverse before reaching the observer. Similarly, photons emitted by observed galaxies are lensed bythe low redshift end of the same large-scale structures. The signal expected from a cross-correlationmeasurement between the two lenses can be related to the fluctuations in their common foregroundmatter field, more precisely by the matter power spectrum P(k,z). The lensing signal is obtained from anextended first-order Limber integration over the past light cone up to the horizon distance χH, weightedby geometrical factors W i(χ), assuming a flat cosmology (Limber, 1954; LoVerde et al., 2008; Kilbingeret al., 2017):CκCMBκgal` =∫ χH0dχW CMB(χ)W gal(χ)P(`+1/2χ;z). (5.1)In the above expression, χ is the comoving distance from the observer, ` is the angular multipole, and zis the redshift. The lensing kernels are given byW i(χ) =3ΩmH202c2χgi(χ)(1+ z), (5.2)withggal(χ) =∫ χHχdχ ′n˜(χ ′)χ ′−χχ ′andgCMB(χ) =[1− χχ∗]H(χ∗−χ). (5.3)Here the constant c is the speed of light in vacuum and χ∗ is the comoving distance to the surface of lastscattering. The term n˜(χ) is related to the redshift distribution of the observed galaxy sources, n(z), byn˜(χ) = n(z)dz/dχ , which depends on the depth of the survey. The Heaviside function H(x) guaranteesthat no contribution comes from beyond the surface of last scattering as the integration in equation 5.1approaches the horizon.The angular cross-spectrum described by equation 5.1 is related to correlation functions in config-uration space, in particular between the CMB lensing map and the tangential shear (Miralda-Escude,1991):ξ κCMBγt(ϑ) =12pi∫ ∞0d` `CκCMBκgal` J2(`ϑ), (5.4)where, J2 is the Bessel function of the first kind of order 2, and the quantity ϑ represents the angularseparation on the sky. Details about measurements of CκCMBκgal` and the tangential shear γt – relevant forequations 5.1 and 5.4 respectively – are provided in Section 5.4.92  0.1-0.3broad0.3-0.5n(z)0.5-0.70.7-0.90 0.5 1 1.5 2> 0.9zFigure 5.1: Redshift distribution of the selected KiDS-450 sources in the tomographic bins (un-normalized), calibrated using the DIR method described in Hildebrandt et al. (2017). Then(z) of the broad zB ∈ [0.1,0.9] bin is shown in black in all panels for reference, while then(z) for the five tomographic bins are shown in red. The mean redshift and effective numberof galaxies in each tomographic bin are summarized in Table 5.1.Our predictions are obtained from the NICAEA6 cosmological tool (Kilbinger et al., 2009), assuminga non-linear power spectrum described by the Takahashi et al. (2012) revision of the HALOFIT model(Smith et al., 2003).5.3 The data sets5.3.1 KiDS-450 lensing dataThe KiDS-450 lensing data that we use for our measurements are based on the third data release ofdedicated KiDS observations from the VLT Survey Telescope at Paranal, in Chile, and are described inKuijken et al. (2015), Hildebrandt et al. (2017) and de Jong (2017 in prep.). These references describethe reduction and analysis pipelines leading to the shear catalogues, and present a rigorous and exten-sive set of systematic verifications. Referring to these papers for more details, we summarise here theproperties of the data that directly affect our measurement.Although the full area of the KiDS survey will consist of two large patches on the celestial equatorand around the South Galactic Pole, the observing strategy was optimized to prioritize the coverage ofthe GAMA fields (Liske et al., 2015). The footprint of the KiDS-450 data is consequently organizedin five fields, G9, G12, G15, G23 and GS, covering a total of 449.7 deg2 While the multiband imagingdata are processed by Astro-WISE (de Jong et al., 2015), the lensing r-band data are processed by theTHELI reduction method described in Erben et al. (2013). Shape measurements are determined using theself-calibrated LENSFIT algorithm (based on Miller et al., 2013) detailed in Fenech Conti et al. (2017).As described in Hildebrandt et al. (2017), each galaxy is assigned a photometric redshift probability6www.cosmostat.org/software/nicaea/93Table 5.1: Summary of the data properties in the different tomographic bins. The effective numberof galaxies assumes the estimation method of Heymans et al. (2012).zB cut z¯ neff (gal/arcmin2) σe[0.1, 0.9] 0.72 7.54 0.28[0.1, 0.3] 0.75 2.23 0.29[0.3, 0.5] 0.59 2.03 0.28[0.5, 0.7] 0.72 1.81 0.27[0.7, 0.9] 0.87 1.49 0.28>0.9 1.27 0.90 0.33distribution provided by the software BPZ (Benı´tez, 2000). The position of the maximum value of thisdistribution, labelled zB, serves only to divide the data into redshift bins. Inspired by the KiDS-450cosmic shear measurement, we split the galaxy sample into five redshift bins: zB ∈ [0.1,0.3], [0.3,0.5],[0.5,0.7], [0.7,0.9] and > 0.9. We also define a broad redshift bin by selecting all galaxies falling in therange zB ∈ [0.1,0.9]. The KiDS-450 cosmic shear measurement did not include the zB > 0.9 bin becausethe photo-z and the shear calibration were poorly constrained there. For this reason, we do not use thisbin in our cosmological analysis either. Instead, we estimate these calibration quantities directly fromour measurement in Section 5.5.7.For each tomographic bin, the estimate of the redshift distribution of our galaxy samples, n(z),is not obtained from the stacked BPZ-PDF, but from a magnitude-weighted scheme (in 4-dimensionalugri magnitude space) of a spectroscopically matched sub-sample. In Hildebrandt et al. (2017), this‘weighted direct calibration’ or ‘DIR’ method was demonstrated to be the most precise covering ourredshift range, among four independent n(z) estimation techniques. Figure 5.1 shows these weightedn(z) distributions, which enter the theoretical predictions through equation 5.1, along with the effectivenumber density per bin. In order to preserve the full description of the data in the high redshift tail,from where most of the signal originates, we do not fit the distributions with analytical functions, aswas done in previous work (Hand et al., 2015; Liu et al., 2015, K16, HD16). Fitting functions tend tocapture well the region where n(z) is maximal, however they attribute almost no weight to the (noisy)high redshift tail. This is of lesser importance in the galaxy lensing auto-correlation measurements, butbecomes highly relevant for the CMB lensing cross-correlation. Instead, we use the actual histogramsin the calculation (as in Liu et al., 2015), recalling that their apparent spikes are smoothed by the lensingkernels in equation 5.3. What is apparent from Fig. 5.1, and of importance for this analysis, is that alltomographic bins have a long tail that significantly overlaps with the CMB lensing kernel, especiallythe first tomographic bin. These tails are caused by inherent properties to the ugri-band photo-z of theKiDS-450 data, and given the wavelength range and SNR, some high-z tails are expected (Hildebrandtet al., 2017). This feature is well captured by the mean redshift distributions, which are listed in Table5.1.Based on the quality of the ellipticity measurement, each galaxy is assigned a LENSFIT weight w,94plus a multiplicative shear calibration factor – often referred to as the m-correction or the shear bias – thatis obtained from image simulations (Fenech Conti et al., 2017). This calibration is accurate to better than1% for objects with zB < 0.9, but the precision quickly degrades at higher redshifts. As recommended,we do not correct for shear bias in each galaxy, but instead compute the average correction for eachtomographic bin (see equation 5.7). In the fifth tomographic bin, we expect to find residual biasesin the m-correction, but apply it nevertheless, describing in Section 5.5.7 how this correction can beself-calibrated. To be absolutely clear, we reiterate that we do not include this fifth bin in our maincosmological analysis. The effective number density and the shape noise in each tomographic bin arealso listed in Table 5.1.Following Hildebrandt et al. (2017), we apply a c-correction by subtracting the weighted meanellipticity in each field and each tomographic bin, but this has no impact on our analysis since this cterm does not correlate with the CMB lensing data.5.3.2 Planck κCMB mapsThe CMB lensing data that enter our measurements are the κCMB map obtained from the 2015 publicdata release,7 thoroughly detailed in Planck Collaboration XV (2016). The map-making procedure isbased on the quadratic estimator described in Okamoto et al. (2003), which is applicable to a suiteof multi-frequency temperature and polarization maps. Frequencies are combined so as to removeforeground contamination, while other sources of secondary signal (mainly emissions from the Galacticplane, from point sources and hot clusters) are masked in the CMB maps, prior to the reconstruction.If some of these are not fully removed from the lensing maps, they will create systematic effects inthe κCMB map that show up differently in the cross-correlation measurement compared to the auto-spectrum analysis. For example, there could be leakage in the CMB map coming from, e.g., residualthermal Sunyaev-Zel’dovich signal that is most likely located near massive clusters. These same clustersare highly efficient at lensing background galaxies, hence our cross-correlation measurement would besensitive to this effect. Indeed, the 〈tSZ× γt〉 correlation, as recently measured in Hojjati et al. (2016),has a very large signal to noise and could possibly be detected in a targeted analysis. Although it isdifficult to assess the exact level of the tSZ signal in our κCMB map, the cleaning made possible fromthe multi-frequency observations from Planck is thorough, reducing the residual contaminants to a verysmall fraction. No quantitative evidence of such leakage has been reported as of yet, and we thereforeignore this in our analysis.Regions from the full sky lensing map that overlap with the five KiDS footprints are extracted,including a 4◦ extension to optimise the signal-to-noise ratio of the measurement (see HD16). ThePlanck release of lensing data also provides the analysis mask, which we apply to the κCMB map priorto carrying out our measurement.87Planck Legacy Archive: pla.esac.esa.int/pla/#cosmology8 This procedure does not entirely capture the masking analysis since the mask was applied on the temperature field, not onthe lensing map. The reconstruction process inevitably leaks some of the masked regions into unmasked area, and vice versa.Applying this mask will therefore only remove the most problematic regions.955.4 The measurementsThis section presents the cross-correlation measurements, which are performed with two independentestimators: ξ κCMBγt (equation 5.4); and the POLSPICE measurement of CκCMBκgal` (equation 5.1). Thesetechniques were used and rigorously validated in previous work and we refer the interested reader toHD16, K16, and references therein for more details. The reasons for conducting our analysis with thesetwo estimators instead of only one are two fold: firstly, they do not probe identical physical scales,which makes them complementary when carried out on surveys covering patchy regions; secondly,being completely independent codes, residual systematics arising from inaccuracies in the analysis couldbe identified through their different effect on these two statistics.5.4.1 The ξ κCMBγt estimationThe first estimator presented in this chapter, ξ κCMBγt , was recently introduced in HD16, and used laterin Singh et al. (2017). It is a full configuration-space measurement that involves minimal manipulationof the data. The calculation simply loops over each pixel of the κCMB maps and defines concentricannuli with different radii ϑ , therein measuring the average tangential component of the shear, γt, fromthe KiDS galaxy shapes. For this reason, it is arguably the cleanest route to performing such a cross-correlation measurement, even though there appears to be a limit to its accuracy at large angles in somecases due to the finite support of the observation window (Mandelbaum et al., 2013). That being said, itnevertheless bypasses a number of potential issues that are encountered with other estimators (see HD16for a discussion). The ξ κCMBγt estimator is given byξ κCMBγt(ϑ) =∑i j κ iCMBei jt wj∆i j(ϑ)∑i j w j∆i j(ϑ)11+K(ϑ), (5.5)where the sum first runs over the κCMB pixels ‘i’, then over all galaxies ‘ j’ found in an annulus of radiusϑ and width ∆, centred on the pixel i. In this local coordinate system, ei jt is the tangential component ofthe LENSFIT ellipticity from the jth galaxy relative to pixel i. The exact binning scheme is described by∆i j(ϑ), the binning operator:∆i j(ϑ) =1, if∣∣θ i−θ j∣∣< ϑ ± ∆2 ,0,otherwise ,(5.6)where θ i and θ j are the observed positions of the pixel i and galaxy j. Following HD16, the bin width∆ is set to 30 arcmin, equally spanning the angular range [1, 181] arcmin with six data points. Largerangular scales capture very little signal with the current level of statistical noise. We verified that ouranalysis results are independent of our choice of binning scheme. In equation 5.5, w j is the LENSFITweight of the galaxy j and K(ϑ) corrects for the shape multiplicative bias m j that must be applied to96−4−2024  0.1−0.9EEEBFid0.1−0.3 0.3−0.50 100 200−4−2024ξκCMB−γt×105ϑ [arcmin]0.5−0.7100 200ϑ [arcmin]0.7−0.9100 200ϑ [arcmin]>0.9−20246  0.1−0.9EEEBFid0.1−0.3 0.3−0.5500 1000 1500−20246CκCMB−κgalℓ×109ℓ0.5−0.7500 1000 1500ℓ0.7−0.9500 1000 1500ℓ>0.9Figure 5.2: Cross-correlation measurement between Planck 2015 κCMB maps and KiDS-450 lens-ing data. The upper part presents results from the ξ κCMBγt estimator, while the lower partshows the estimation of CκCMBκgal` . Different panels show the results in different tomographicbins, with predictions (solid curve) given by equations 5.1 and 5.4 in our fiducial cosmology.The black squares show the signal, whereas the red circles present the EB null test describedin Section 5.5.2, slightly shifted horizontally to improve the clarity in this figure. The errorbars are computed from 100 CMB lensing simulations.97the lensing data (Fenech Conti et al., 2017):11+K(ϑ)=∑i j w j∆i j(ϑ)∑i j w j(1+m j)∆i j(ϑ). (5.7)The theoretical predictions for ξ κCMBγt are provided by equation 5.4. We apply the same binning aswith the data, averaging the continuous theory lines inside each angular bin. We show in the upper panelof Fig. 5.2 the measurements in all tomographic bins, compared to theoretical predictions given by ourfiducial WMAP9+BAO+SN cosmology. The estimation of our error bars is described in Section 5.4.3.We also project the galaxy shape components onto e×, which is rotated by 45◦ compared to et. Thiseffectively constitutes a nulling operation that can inform us of systematic leakage, in analogy to the EBtest performed in the context of cosmic shear. For this reason, we loosely refer to ‘EE’ and ‘EB’ testsin this chapter, when we are in fact comparing κCMB× et and κCMB× e×, respectively. We note thatthe past literature referred to such an EB measurement as the ‘B-mode test’, which can be misleadingfor the non-expert. Indeed, the proper B-mode test refers to the BB measurement in weak lensinganalyses, a non-lensing signal that can be caused by astrophysics and systematics. The EB signal testasserts something more fundamental: since B changes sign under parity, and E does not, a non-zero EBmeans a violation of the parity of the shear/ellipticity field (Schneider, 2003). That is not expected fromlensing alone, so could only come from a systematic effect that does not vanish under averaging. OurEB measurement is shown with the red symbols in Fig. 5.2. We find by visual inspection that in mosttomographic bins, these seem closely centred on zero, but not in all cases. To quantify the significanceof this EB measurement, we estimate the confidence level with which these red points deviate from zero.We detail in Section 5.5.2 how we carry out that test and show that they are consistent with noise.We have carried out an additional null test presented in HD16, which consists in randomly rotatingthe shapes of the galaxies before the measurement (κCMB× random). This test is sensitive to the noiselevels in the galaxy lensing data and hence affected by the shape noise σe listed in Table 5.1. We findthat the resulting signal is fully consistent with zero in all tomographic bins.5.4.2 The CκCMBκgal` estimationThe second estimator uses the same data as our ξ κCMBγt analysis, namely the κCMB map and the KiDSshear catalogues, but requires additional operations on the data, including harmonic space transforms.This is accomplished with the POLSPICE numerical code (Szapudi et al., 2001a; Chon et al., 2004) run-ning in polarization mode, where the {T, Q, U} triplets are replaced by {κCMB,0,0} and {0,−e1,e2}.The code first computes the pseudo-C` of the maps and of the masks, then transforms the results intoconfiguration space quantities, that are finally combined and transformed back into Fourier space. Theoutput of POLSPICE is therefore an estimate of the cross-spectrum CκCMBκgal` . While POLSPICE is fre-quently used for CMB analyses, it was applied for the first time in the context of CMB lensing ×galaxy lensing by K16 and serves as a good comparison to the configuration-space estimator describedin Section 5.4.1. One main advantage of this estimator is that in principle different ` bands are largelyuncorrelated, which makes the covariance matrix almost diagonal and hence easier to estimate.98The POLSPICE measurement9 is presented in the lower panel of Fig. 5.2, plotted against the theo-retical predictions given by equation 5.1. The EB data points are directly obtained from the tempera-ture/B-mode output provided by the polarization version of the code, and are further discussed in Section5.5.2.Note that our choice of the γt and POLSPICE estimators was motivated by our desire to avoid pro-ducing κgal maps in order to reduce the risks of errors and systematic biases that can arise in the mapmaking stage in the presence of a mask as inhomogeneous as that of the KiDS-450 data. These twoestimators produce correlated measurements, but they probe different scales. The γt estimator is accu-rate at the few percent level, as verified on full mock data in HD16, and the POLSPICE code has beenthoroughly verified and validated on the same mocks as well. We refer the reader to K16 and HD16 fordetails of these tests.5.4.3 Covariance estimationThe κCMB map reconstructed from the Planck data is noise dominated for most Fourier modes (PlanckCollaboration XV, 2016). It is only by combining the full sky temperature and polarization maps thatthe Planck Collaboration could achieve a lensing detection of 40σ .Since the noise NCMB is larger than the signal κCMB at every scale included in our analysis (HD16),we can evaluate the covariance matrix from cross-correlation measurements between the 100 Plancksimulated lensing maps (also provided in their 2015 public data release) and the tomographic KiDSdata:CovκCMBκgal``′ ' 〈∆CˆNCMBκgal` ∆CˆNCMBκgal`′ 〉 ; (5.8)andCovκCMBγtϑϑ ′ ' 〈∆ξˆNCMBγtϑ ∆ξˆNCMBγtϑ ′ 〉 , (5.9)where the ‘hats’ refer to measured quantities, ∆xˆ = xˆ− x¯, and the brackets represent the average overthe 100 realizations. This method assumes that the covariance is completely dominated by the CMBlensing and neglects the contribution from the shear covariance. This is justified by the fact that thesignal from the former is about an order of magnitude larger, and hence completely drives the statisticaluncertainty (HD16). The error bars shown in Fig. 5.2 are obtained from these matrices (from the squareroot of the diagonals). For each tomographic bin, the CovκCMBκgal``′ matrix has 25 elements, whereas theCovκCMBγtϑϑ ′ matrix has 36. The 100 realizations are enough to invert these matrices one at a time with acontrollable level of noise bias, and the numerical convergence on this inverse is guaranteed (Lu et al.,2010). Note that this strategy fails to capture the correlation between tomographic bins, which are notrequired by our cosmological analysis presented in Section 5.5.6. If needed in future analyses, thesecould be estimated from full light-cone mock simulations.9POLSPICE has adjustable internal parameters, and we use THETAMAX = 60◦, APODIZESIGMA = 60◦ and NLMAX =3000.99For both estimators, the covariance matrix is dominated by its diagonal, with most off-diagonalelements of the cross-correlation coefficient matrix being under ±10%. Some elements reach largervalues, ±40% correlation at the most, but these are isolated, not common to all tomographic bins, andare consistent with being noise fluctuations, given that we are measuring many elements from ‘only’100 simulations. This partly explains why our cosmological results are not based on a joint tomographicanalysis. We keep the full matrices in the analysis, even though we could, in principle, include only thediagonal part in the POLSPICE measurement. Nevertheless, we have checked that our final results areonly negligibly modified if we use this approximation in the χ2 calculation, suggesting that one couldreliably use a Gaussian approximation to the error estimation in this type of measurement (see equation23 in HD16).5.5 Cosmological inferenceGiven the relatively low signal-to-noise ratio of our measurement (Fig. 5.2), we do not fit our signalfor the six parameter ΛCDM cosmological model. Instead, we follow the strategy adopted in earlierstudies: we compare the measured signal to our fiducial cosmological predictions, treating the normal-ization as a free parameter ‘A’. If the assumed fiducial cosmology is correct and in absence of othersystematic effects, A is expected to be consistent with unity. As discussed in previous studies, A isaffected by a number of effects that can similarly modulate the overall amplitude of the signal. Asidefrom its sensitivity to cosmology – our primary science target – this rescaling term will absorb con-tributions from residual systematic errors in the estimation of n(z), from mis-modelling of the galaxyintrinsic alignments, from residual systematic bias in the shear multiplicative term m (equation 5.7),from astrophysical phenomena such as massive neutrinos and/or baryonic feedback, and from residualsystematics in the cross-correlation estimators themselves (K16 and HD16).In this section, we first present our constraints on A; we then quantify how the different effectslisted above can impact our measurements, and finally present our cosmological interpretation. Ourprimary results assume the fiducial WMAP9+BAO+SN cosmology, i.e. we first place constraints onAfid, however we also report constraints on AKiDS and APlanck, obtained by assuming different baselinecosmologies.5.5.1 SignificanceTo measure A, we first compute the χ2 statistic:χ2 = ∆xT Cov−1 ∆x , (5.10)with∆x = ξˆ κCMBγt−Aξ κCMBγt or ∆x = CˆκCMBκgal−ACκCMBκgal (5.11)100for the configuration-space and POLSPICE estimators, respectively. As before, quantities with ‘hats’are measured, and the predictions assume the fiducial cosmology, unless stated otherwise. The signal-to-noise ratio (SNR) is given by the likelihood ratio test, which measures the confidence at which wecan reject the null hypothesis (i.e., that there is no signal, simply noise) in favour of an alternativehypothesis described by our theoretical model with a single parameter A (see Hojjati et al., 2016, for arecent derivation in a similar context). We can write SNR =√χ2null−χ2min, where χ2null is computed bysetting A = 0, and χ2min corresponds to the best-fit value for A. The error on A is obtained by varying thevalue of A until χ2A−χ2min = 1 (see, e.g. Wall et al., 2003).We include two additional statistical corrections to this calculation. The first is a correction factorthat multiplies the inverse covariance matrix, α = (Nsim−Nbin− 2)/(Nsim− 1) = 0.94, to account forbiases inherent to matrix inversion in the presence of noise (Hartlap et al., 2007). Here Nbin is the numberof data bins (five for CκCMBκgal and six for ξ κCMBγt) and Nsim is the number of simulations (100) used inthe covariance estimation. There exists an improved version of this calculation based on assuming at-distribution in the likelihood, however with our values of Nbin and Nsim, the differences in the invertedmatrix would be of order 10–20% (Sellentin et al., 2016), a correction on the error that we ignore giventhe relatively high level of noise in our measurement.The second correction was first used in HD16, and consists of an additional error on A due tothe propagated uncertainty coming from the noise in the covariance matrix (Taylor et al., 2014). Thiseffectively maps σA→ σA(1+ε/2), where ε =√2/Nsim+2(Nbin/N2sim) = 0.145. These two correctionfactors are included in the analysis. The results from our statistical investigation are reported in Table5.2, where we list χ2min, χ2null, SNR and A for every tomographic bin. The theoretical predictions providea good fit to the data given that for our number of degrees of freedom ν = Nbin−1, ν−√2ν < χ2min <ν +√2ν . In other words, all our measured χ2 fall within the expected 1σ error. We also compute thep-value for all these χ2 measurements at the best-fitting A in order to estimate the confidence at whichwe can accept or reject the assumed model. Assuming Gaussian statistics, p-values smaller than 0.01correspond to a 99% confidence in the rejection of the model (the null hypothesis) by the data, and areconsidered ‘problematic’. Our measured p-values, also listed in Table 5.2, are always larger than 0.12,meaning that the model provides a good fit to the data in all cases.These tomographic measurements are re-grouped in Fig. 5.3, where we compare the redshift evolu-tion of A for both estimators. We mark the 1σ region of the broad bin n(z)with the solid horizontal lines,and see that all points overlap with this region within 1σ . This is an indication that the relative growthof structure between the tomographic bins is consistent with the assumed ΛCDM model. For the broadn(z), the signal prefers an amplitude that is 23−31 percent lower than the fiducial cosmology, i.e., the1σ region shown by the horizontal solid lines in Fig. 5.3 is offset from unity by that amount. The maincosmological result that we quote from the zB ∈ [0.1,0.9] measurement is that of the γt estimator due toits higher SNR, as seen from comparing the top two rows of Table 5.2. For our fiducial cosmology, wefindAfid = 0.69±0.15 . (5.12)101Table 5.2: Summary of χ2, SNR and p-values obtained with the two different pipelines. TheCκCMBκgal` measurements have 4 degrees of freedom (5 ` bins − 1 free parameter), whereas theconfiguration space counterpart ξ κCMBγt(ϑ) has one more, with 6 ϑ -bins. Afid is the best-fitamplitude that scales the theoretical signals in the fiducial cosmology, according to equation5.11, also shown in Fig. 5.3. The numbers listed here include the covariance debiasing factorα and the extra error ε due to the noise in the covariance (see main text of Section 5.5.1 formore details).zB Estimator χ2min χ2null SNR p-values Afid[0.1, 0.9]CκCMBκgal` 2.80 18.21 3.93 0.53 0.77 ± 0.19ξ κCMBγt 2.88 22.94 4.48 0.64 0.69 ± 0.15[0.1, 0.3]CκCMBκgal` 5.48 8.89 1.85 0.20 0.55 ± 0.30ξ κCMBγt 7.93 13.38 2.34 0.12 0.53 ± 0.24[0.3, 0.5]CκCMBκgal` 2.95 4.95 1.42 0.50 0.71 ± 0.51ξ κCMBγt 1.44 4.19 1.66 0.84 0.60 ± 0.37[0.5, 0.7]CκCMBκgal` 4.00 10.13 2.47 0.35 0.87 ± 0.35ξ κCMBγt 2.00 6.45 2.11 0.77 0.55 ± 0.26[0.7, 0.9]CκCMBκgal` 5.12 10.04 2.22 0.23 0.79 ± 0.36ξ κCMBγt 2.78 15.41 3.55 0.65 1.02 ± 0.29> 0.9CκCMBκgal` 4.70 12.92 2.87 0.26 0.83 ± 0.29ξ κCMBγt 4.68 22.64 4.24 0.38 0.95 ± 0.22Varying the cosmology to the best fit KiDS-450 and Planck cosmologies10,11,12, we obtainAKiDS = 0.86±0.19 and APlanck = 0.68±0.15 . (5.13)The relative impact of these different cosmologies on our signal is presented in Fig. 5.4, where we seethat the KiDS-450 cosmology prediction mostly differ from the other two at large scales. The signalfrom our fiducial cosmology agrees with that assuming the best-fit Planck cosmology to better than 5%in all tomographic bins.Figure 5.3 demonstrates there are small differences between the two estimators at fixed cosmology,especially at high redshift. As mentioned in Section 5.4.2, the scales being probed are not identical, andtherefore some differences in the recovered values of A are expected. Nevertheless, within the currentstatistical accuracy, the two estimators are fully consistent with one another.Visually, the ξ κCMBγt(ϑ) estimator seems to show a mild trend for decreasing values of A in lowerredshift bins. Although such an effect could point towards a number of interesting phenomena sup-pressing power for source galaxies at z . 0.7 (e.g., modification to the growth history compared to10Fiducial (Ωm,ΩΛ,Ωb,σ8,h,ns) = (0.29,0.71,0.047,0.83,0.69,0.97).11KiDS-450 (Ωm,ΩΛ,Ωb,σ8,h,ns) = (0.25,0.75,0.047,0.85,0.75,1.09).12Planck (Ωm,ΩΛ,Ωb,σ8,h,ns) = (0.32,0.68,0.049,0.80,0.67,0.97).1020.75 0.59 0.72 0.87 1.2700.511.5ξκ CMBγt Az¯0.1−0.3 0.3−0.5 0.5−0.7 0.7−0.9 >0.900.511.5HT f redCκ CMBκ galZB binAFigure 5.3: Tomographic measurement of Afid, defined in equation 5.11, assuming our fiducialcosmology. The two panels present results from the two cross-correlation estimators (labelledin the top left corner). Black symbols assume no IA, while colour symbols include correctionfactors from two IA models ( fred in magenta and HT in blue, see Sec. 5.5.3). The horizontalsolid lines of a given colour enclose the 1σ region measured in the broad zB ∈ [0.1,0.9] bin,while the dotted horizontal lines indicate the fiducial values (Afid = 1). The mean sourceredshift in each bin is indicated at the top and summarized in Table 5.1. The mean in thefirst bin is high because of the long tail, visible in Fig. 5.1. The best-fit values in differentcosmologies are Afid = 0.69±0.15, AKiDS = 0.86±0.19 and APlanck = 0.68±0.15.500 1000 1500 2000 2500 3000−0.2−0.15−0.1−0.0500.05ℓCℓ/Cfidℓ−1  KiDS450Planck0.1−0.30.3−0.50.5−0.70.7−0.9>0.9Figure 5.4: Fractional effect on the signal when changing the fiducial cosmology to Planck orKiDS-450. Different symbols show the impact in different tomographic bins, relative to thefiducial predictions. Current measurements are limited to ` < 2000.103Table 5.3: p-values for the EB test obtained for the 6 tomographic bins. Highlighted in bold is thep-value ≤ 0.01. The column labelled C10 refers to POLSPICE measurements in which the dataare organized in 10 bins instead of five. These calculations assume t-distributed likelihoods(following Sellentin et al., 2016).zB Ct−dist Ct−dist10 ξt−dist[0.1, 0.9] 0.20 0.09 0.58[0.1, 0.3] 0.01 0.09 0.52[0.3, 0.5] 0.39 0.63 0.08[0.5, 0.7] 0.94 0.57 0.78[0.7, 0.9] 0.21 0.20 0.16> 0.9 0.53 0.54 0.68the fiducial cosmology, additional feedback processes from baryons or massive neutrinos, or redshift-dependent contamination from IA) the significance of this redshift dependence is too low to draw anyrobust conclusions.What is significantly seen from Fig. 5.3 is that the signal is generally low compared to the fiducialand Planck cosmologies. Our measurements of the amplitude A prefer instead the KiDS-450 cosmology,which also aligns with the CFHTLenS cosmic shear results (Kilbinger et al., 2013). We further quantifythis comparison in Section 5.5.6, first presenting results from our set of null tests, and then examiningthree sources of contamination and systematic biases that potentially affect our signal. In this work,we neglect the effect of source-lens coupling (Bernardeau, 1998), which could possibly act as anothersecondary signal, biasing the signal low. As it is the case for cosmic shear, this effect should be toosmall ( < 10%) to affect our results significantly, and further investigation will be required to interpretcorrectly the measurements from future surveys.5.5.2 Null testsWe have shown in Section 5.4 and in Fig. 5.2 (red circles) that the parity violation EB test seemsconsistent with noise in most tomographic bins, but occasionally this is not obvious. In this section,we investigate the significance of these measurements. Statistically, this is accomplished by measuringthe confidence at which we can reject the null hypothesis ‘parity is not violated’. We therefore re-runthe full χ2 statistical analysis13 and measure the p-value about the model with A = 0. Low p-valuescorrespond to high confidence of rejection, i.e., that some residual systematic effect might be causingand apparent parity violation. This type of measurement strongly probes the tail of the χ2 distribution,hence assuming a Gaussian likelihood would provide inaccurate estimations of the p-values, even whenincluding the Hartlap et al. (2007) debiasing α factor. Instead, we follow Sellentin et al. (2016) andassume a t-distribution for the likelihood, which better models the tail of the likelihood. Table 5.3 lists13Due to the absence of parameters in the null hypothesis, the EB case has one additional degree of freedom compared tothe EE case.104all these p-values, highlighting in bold one that seems slightly problematic (p-value ≤ 0.01). Since thissingle low p-value is only seen in one of the two estimators, we conclude that it must originate fromexpected noise fluctuations, and not from the data themselves. This conclusion is additionally supportedby the fact that the level of B-modes in the KiDS data (i.e., the BB measurement) is consistent with zeroon the scales we are probing (Hildebrandt et al., 2017).For the sake of testing the robustness of the EB POLSPICE measurement, we have additionallyinvestigated the effect of changing the number of bins from 5 to 10. In the EE case, the recoveredvalues of A and the SNR are similar to those presented in Table 5.2, from which we conclude thatthis comes with no gain. However, when applied to the EB null test, something interesting happens:the ‘problematic’ measurement (p-value = 0.01 in the zB ∈ [0.1,0.3] bin, Table 5.3) relaxes to 0.09, asseen in the column labelled C10. This is another indication that the cause of the low p-value originatesfrom fluctuations in the noise – which is affected in the binning process – without pointing to residualsystematic effects in the data.We have verified that our measurement of A is robust against the removal of some scales. When weexclude the largest or the smallest angular bin in the ξ κCMBγt measurement, results change by at most0.7σ , generally by less than 0.2σ . This gives us confidence in the robustness of our measurement. Thesame holds when removing the highest ` bin from the POLSPICE measurement, but not for the lowest` bin, which captures the peak of the signal, and therefore contributes significantly to the SNR. At thesame time, this test illustrates that we are currently not sensitive to the effect of massive neutrinos norto baryonic feedback, which mainly affect these non-linear scales.5.5.3 Effect of intrinsic alignmentsIntrinsic alignments (IA) are a known secondary effect in the cross-correlation of galaxy lensing andCMB lensing that lowers the amplitude of the measured signal (Hall et al., 2014; Troxel et al., 2014;Chisari et al., 2015). It is therefore important to investigate how much IA could contribute to the ob-served low values of A reported in Table 5.2. To estimate the contamination level, we compare twodifferent models, which we then apply equally to both estimators, CκCMBκgal` and ξκCMBγt(ϑ).First, we follow Hall et al. (2014, ‘HT-IA’ model hereafter) in using the ‘linear non-linear alignment’model of Bridle et al. (2007) with the SuperCOSMOS normalization found in Brown et al. (2002). Werecall that this prescription comes from constraints at z = 0.1 that are independent of galaxy type orcolour, and that the effect of IA in this model is to reduce the amplitude of the observed signal, as thegalaxies tend to align radially towards each other. The scale-dependence of the alignment contributionis similar to the lensing signal, as seen in Hall et al. (2014) and in Fig. 5.5, hence we only quote thepercentage of contamination at `= 1000 for reference. This also allows us to confidently use the sameIA contamination levels for the configuration space estimator, since rescaling CκCMBκgal` by a constantrescales ξ κCMBγt by the same constant (as per equation 5.1). For each of the five redshift bins consideredin this chapter, starting from the lower redshift, we estimate a {10,17,10,8,5}% contamination to thesignal, respectively. For the broader tomographic bin zB ∈ [0.1,0.9], we estimate an 11% contamination.In other words, within the HT-IA model, the measured value of Afid in the broad bin (equation 5.12)105−0.15−0.1−0.0500.050.1CIA−fredℓ/Cfidℓ  200 400 600 800 1000 1200 1400 1600 1800 2000−0.2−0.100.1ℓCIA−HTℓ/Cfidℓ  0.7−0.9>0.90.1−0.90.1−0.30.3−0.50.5−0.7Figure 5.5: Strength of the contamination by intrinsic galaxy alignments for different tomographicbins, assuming our fiducial cosmology and the linear non-linear alignment model. The dif-ference between lines is caused by changes in n(z) (and in the red fraction in the fred-IAmodel).should be corrected toAHTfid = 0.77±0.15 . (5.14)The error bars here are not modified compared to the no-IA case since this contamination signal isadditive. This model is the simplest as it assumes no luminosity or redshift dependence of the alignmentnormalization, and adopts the same alignment prescription for all galaxies regardless of morphologicaltype or colour.Second, we estimate the contamination from the alignment model of Chisari et al. (2015, ‘ fred-IA’ model hereafter) that allows for differential contributions based on galaxy colour/morphology. Weassume that blue galaxies do not contribute at all, consistent with observations (Heymans et al., 2013;Mandelbaum et al., 2011), even though this null measurement remains poorly constrained. We estimatethe red fraction directly from the data in each redshift bin using the best-fit spectral template returnedby BPZ for each source, referred to as TB. Motivated by Simon et al. (2015), we identify red galaxiesas objects with TB < 1.5.For the five tomographic bins, we obtain fractions of red galaxies fred ={0.04,0.12,0.27,0.18,0.04}; we estimate fred = 0.15 for the broad bin. We then use the align-ment amplitude for the red galaxies from Heymans et al. (2013) to obtain an estimate of alignmentcontamination given our red fractions. These results are presented in the upper panel of Fig. 5.5.With this method, we estimate a {2,11,14,7,1}% contamination from intrinsic alignments in the106tomographic bins, and a 9% contamination in the zB ∈ [0.1,0.9] bin.14 Then we can estimateA fredfid = 0.75±0.15 . (5.15)One caveat with this model is that the K-correction and evolutionary corrections are uncertain at highredshift, which could result in biased estimates of the red fraction (see discussion in Chisari et al.,2015). This has an impact on the exact level of intrinsic alignment contamination by red galaxies, butwe neglect this effect in this work.Both methods are broadly consistent even though they differ in details, especially in the lowestredshift bin. For instance, the second method captures the redshift differences observed in the dataand takes into account the split in contributions arising from different galaxy types, which introducesa slightly different redshift dependence of the IA signal. Nevertheless, the overall trends between theHT and the fred-IA models are similar, although that is not the case for all IA models (see, for example,the tidal torque theory from Codis et al., 2015, in which the sign of the IA effect on the signal is theopposite). There remains a large uncertainty in the modelling of the IA contamination, and we do notknow which model, if any, should enter in our cosmological interpretation.According to these estimations, both the HT-IA and fred-IA models help to bring A closer to unity.From the contamination levels listed above, at most 17% of the observed cross-correlation signal canbe canceled by IA contamination in our tomographic bins. After correcting for this effect in eachtomographic bin, most points agree with Afid = 1 within 1σ . This is shown with the colour symbols inFig. 5.3.Finally, we note that the uncertainty on the level of IA contamination quoted in the section is high,especially because of the unknown signal from the blue galaxies. For instance, at the 1σ level andassuming the linear non-linear alignment model, the IA contamination from blue galaxies could rangefrom −{10,15,8,6,4}% to +{6,9,5,4,3}% in each tomographic bin, and from −10% to +6% in thezB ∈ [0.1,0.9] bin.5.5.4 Effect of n(z) errorsWe investigate here the impact on our measurement of A from the uncertainty on the source redshiftdistribution. This is estimated from 100 bootstrap resamplings of the source catalogue, as detailed inHildebrandt et al. (2017, the DIR method described therein). These samples consist of internal fluctua-tions in the n(z), which we turn into fluctuations in the signal using equations 5.1–5.3. We present in thetop panel of Fig. 5.6 the fractional error on the signal, i.e. σbootC` /C`. According to this error estimate,the uncertainty on n(z) is up to 8% of the signal in the first redshift bin, then 4, 2 and 1% for the others,and about 3% for the zB ∈ [0.1, 0.9] bin.Note that this quantity is a measure of how the DIR n(z) varies – and how it impacts the signal –across subsamples of the re-weighted spectroscopically-matched catalogue. This catalogue is by itself14 We measured the field-to-field variance in fred and observed that it hardly varies except in the highest redshift bin, wherethe scatter could turn the 1% IA contamination into a 0.0–2.5% contamination. This remains small and should have a negligibleimpact on our results.1070 500 1000 1500 200000.ℓσbootC/Cℓ  0.1−0.30.3−0.50.5−0.70.7−0.90.1−0.9Figure 5.6: Fractional effect on the CκCMBκgal` signal when varying the n(z) between 100 bootstrapresamplings, for the four tomographic bins with zB < 0.9. Shown here is the 1σ scatterdivided by the signal.subsampling the full KiDS sources, and hence is subject to sampling variance. It is therefore likely thatthe error quoted above slightly under-estimates the true error on the signal due to the n(z), as discussedin Section C3.1 of Hildebrandt et al. (2017). However, this is sub-dominant compared to our statisticaluncertainty and is therefore not expected to affect our results.For comparison purposes, we also investigated estimates of the redshift distribution determinedusing the cross-correlation between spectroscopic and photometric samples (known as the CC methodin Hildebrandt et al., 2017; Morrison et al., 2017). This scheme has a high level of noise compared to thefiducial DIR method and we find that the error on the recovered C` in our analysis increases from ∼5%in the DIR case to ∼30% in the CC case. From this we can draw the same conclusion as the KiDS-450cosmic shear analysis, namely that determining the redshift distribution using the cross-correlation CCmethod will remove any discrepancy with a Planck cosmology through inflation of the error bars. Webelieve, however, that the error on the CC estimate is not representative of our actual knowledge of then(z) in the KiDS data, and refer instead to the redshift distribution defined using the DIR method in therest of this chapter.Precision on the KiDS source redshift distribution will soon increase thanks to the ongoing process-ing of near-IR VIKING data (Hildebrandt et al. in prep.), which primarily impacts the high redshifttail so crucial to our measurement. Finally note that in the DIR method we are using a calibrated n(z),estimated from weighted spectroscopic data, hence we do not have to worry nearly as much about catas-trophic photo-z outliers. This was not the case for the analysis presented in HD16, which showed thatfor n(z) estimated directly from photometric data (for e.g., CFHTLenS and RCSLenS), these can easilydominate the error budget, with systematic effects on the signal of the order of 15%. If our measurementcontains more high-redshift objects than our n(z) suggests, our predictions are too low; correcting forthis would lower A.108101 102 1030.ℓ∆CκCMBκgalℓ/CDM−ONLYℓ  zs=0.5 zs=1.5S030.4eV0.05eVDM−onlyAGNMνAGN+MνFigure 5.7: Fractional effect of the AGN baryon feedback and massive neutrinos on the cross-spectra, for different combinations of source planes. The red solid line shows the combinedeffect on the cross-spectrum for sources placed on a single plane at zs = 0.5. The effect of0.05 eV massive neutrinos and AGN feedback are shown separately by the upper dashed anddotted lines (also in red). The lower dashed red line shows the impact of 0.4 eV neutrinos.Blue lines show the same quantities, but for sources placed at zs = 1.5. The dashed black lineshows the ratio between the predictions from Smith et al. (2003) and that of Takahashi et al.(2012).5.5.5 Baryon feedback, massive neutrinos and non-linear modellingAs shown in HD16, baryonic feedback and massive neutrinos can cause an important decrease of thecross-correlation signal, which would translate into lower values of A when compared to a fiducial darkmatter only cosmology. To investigate how this could affect our cosmological results, we modify theP(k,z) term in equation 5.1 to include ‘massive neutrino bias’ and ‘baryon feedback bias’, as detailedin Harnois-De´raps et al. (2015b). The baryon bias was extracted from the OWL simulations, assumingthe AGN model (van Daalen et al., 2011), while the neutrino bias was extracted from the recalibratedHALOFIT code (Takahashi et al., 2012) with total neutrino masses Mν = 0.05, 0.2, 0.4 and 0.6 eV. Ourresults are presented in Fig. 5.7 for two simplified cases, in which the source galaxy populations areplaced on single planes at zs = 0.5 (in red) and at zs = 1.5 (in blue). The figure focuses on the 0.05eV scenario, showing the suppression of power caused by massive neutrinos (3.5% effect on A for bothzs planes, averaged over the ` modes that we measured), by baryonic feedback (5.0% for zs = 1.5 and10.6% for zs = 0.5), and by the combination of both (8.2% and 13.7% for zs = 1.5 and 0.5 respectively).The reason why the effect of baryons is larger on the lower redshift source plane is simply a projectioneffect: the same physical scales subtend different angles on the sky, which contribute differently to ourmeasurement restricted to the ` ∈ [20−2000] range. We also show the effect of 0.4 eV neutrinos (28%and 30% in the two zs slices), which demonstrates a scaling of 7% per 0.1 eV. We note that Mead et al.(2015) proposed an alternative method to account simultaneously for baryons and neutrinos based onthe halo model, which might prove useful in future work.109These two effects contribute at some level to the measurement of A, but it is too early to put con-straints on them based on our measurement. Firstly, the cosmology is not guaranteed to be that of Planck,secondly the exact feedback mechanism that is at play in the Universe remains largely unknown, andthirdly other effects (e.g., IA contamination or errors in n(z)) could explain why our measured Afid islow. However, if the fiducial cosmology is correct and if the intrinsic alignments are well described bythe HT model outlined in Section 5.5.3, then Afid would be brought to unity with Mν = 0.33 ± 0.22 eVin the absence of baryonic feedback, and Mν = 0.19 ± 0.22 eV within the AGN model.We have verified that the uncertainty in the non-linear modelling does not affect our measurement ofA significantly. This is mainly because the angles and redshifts probed by our measurement correspondto scales that are mostly in the linear and mildly non-linear regimes. Replacing the non-linear powerspectrum from the Takahashi et al. (2012) model with that of Smith et al. (2003), corresponding to aradical change in the non-linear predictions at small scales shown in Fig. 5.7 (black dashed line), affectsour measurement of A by only 1–2%. This is well within the statistical uncertainty and can be safelyneglected. Figure 5.7 shows that there is a clear degeneracy between differences in the two models,and the effects of baryonic feedback. However, the Smith et al. (2003) predictions are known to sufferfrom a significant loss of power at small scales, visible in Fig. 5.7, and the state-of-the-art precision onthe non-linear power spectrum, from e.g., the Cosmic Emulator (Heitmann et al., 2014) deviates fromthe Takahashi et al. (2012) model by less than 5% (Mead et al., 2015). This alleviates the degeneracybetween modelling and baryonic feedback effects and further supports our (model-dependent) neutrinomass constraints presented above.5.5.6 Cosmology from broad n(z)In this section, we investigate how our cross-correlation measurement can constrain cosmology, andspecifically compute confidence regions in the (σ8,Ωm) plane. For this calculation we assume masslessneutrinos and no baryonic feedback; we also ignore the error on n(z), but examine our results for thetwo IA models (as well as the ‘no-IA’ case) described in Section 5.5.3.It was shown in Liu et al. (2015) that the amplitude of the cross-correlation signal scales approxi-mately with σ28Ω−0.5m at large, linear scales (` < few hundred), and as σ38Ω1.3m at small scales (` > 1000).Most of our constraints come from small scales, but our measurement includes some large modes downto ` ≈ 200. For this reason, we strike a compromise: we keep the Ω1.3m dependence, as suggested byLiu et al. (2015), but use a σ2.58 dependence, to capture the gradual transition between both. Futuremeasurements will require Monte Carlo algorithms to be run to better capture these dependencies, butthis is unnecessary in our case given the relatively large uncertainty on A.As discussed before, we use the ξ κCMBγt results in the broad zB ∈ [0.1,0.9] tomographic bin becauseit has the highest signal to noise; however, our results would not change significantly if we used theCκCMBκgal` measurement instead. We could also have used the tomographic results, i.e., the A(z) variesin the four bins. However, these measurements are all correlated, probing common low-redshift lenses.Such an approach would require us to calculate and include cross-correlation coefficients between thedifferent tomographic bins when solving for the best-fit cosmology. These could be evaluated from110Figure 5.8: Constraints on σ8 andΩm as estimated from the cross-correlation measurement, ignor-ing potential contamination by intrinsic galaxy alignments (shown in black). The solid lineshows the best fit, while the dashed and dotted lines indicate the 68% and 95% confidencelevel (CL) regions, respectively. The cross-correlation results can be compared to KiDS-450 (green, where IA effects are accounted for), Planck (orange), and WMAP9+SPT+ACT(blue).Figure 5.9: Same as Fig. 5.8, but here assuming 10% contamination from IA in the cross-correlation measurement (equation 5.17), consistent with both the ‘HT-IA’ and the ‘ fred-IA’models.111mock data, but this is not required when working with a single data point for A. Combining this scalingrelation with equations 5.12–5.14, we obtainA = Afid( σ80.831)2.5( Ωm0.2905)1.3= 0.69±0.15 (5.16)andA = AHTfid( σ80.831)2.5( Ωm0.2905)1.3= 0.77±0.15, (5.17)which we use to propagate the error on A into confidence regions in the (σ8,Ωm) plane. We show inFigs. 5.8 and 5.9 how these constraints compare to the results from KiDS-450 cosmic shear (with IA),Planck and pre-Planck experiments.15 Our cross-correlation measurement has a larger overlap with theKiDS-450 constraints, but is still consistent with the Planck cosmology in the sense that their . 95%confidence regions overlap. Including IA reduces the offset from Planck.Given that our signal has different dependences on cosmological (e.g., Ωm and σ8) and nuisance(e.g., m, n(z)) parameters, we can see how this can provide new insights into resolving tensions betweenthe cosmic shear and CMB measurements. For example, whereas the KiDS-450 and CFHTLenS cosmicshear results scale as (1+m)2n2(z), our KiDS-450× Planck lensing measurement scales as (1+m)n(z).This difference could therefore allow us to break the degeneracy in a joint probe analysis. Also note thatin general, we should not exclude the possibility that there could be residual systematics left over in aCMB temperature and polarization analysis — driving the cosmology to higher (σ8,Ωm) values — thatdo not make their way to the CMB lensing map or into the joint probes measurement, in analogy withthe cosmic shear c-term. This is certainly the case for the additive shear bias (the c-correction) describedin Kuijken et al. (2015). Having this new kind of handle can help to identify the cause of disagreementsbetween different probes, and will be central to the cosmological analyses of future surveys. We explorefurther how cross-correlation analyses can be turned into a calibration tool in the next section.5.5.7 Application: photo-z and m-calibrationThe CMB lensing–galaxy lensing cross-correlation signal has been identified as a promising alternativeto calibrate cosmic shear data without relying completely on image simulations (Das et al., 2013; Liuet al., 2016; Schaan et al., 2017). This statement relies on the fact that A absorbs all phenomena thataffect the amplitude of the measurement, i.e., cosmology, intrinsic alignment, n(z), and shear calibration,and that we can marginalize over some of these in order to solve for others.Most of the attention so far has been directed towards the multiplicative term in the cosmic shearcalibration — the m j factor in equation 5.7 — which has an important impact on the cosmologicalinterpretation. In the case of the KiDS-450 data, the shear calibration is known at the percent level15The MCMC chains entering these contour plots can be found on the KiDS-450 website: kids.strw.leidenuniv.nl/cosmicshear2016.php Note also that the WMAP9+SPT+ACT cosmology presented in Figs. 5.8 and 5.9 differ from the fiducialWMAP9+BAO+SN cosmology.112from image simulations for objects with zB < 0.9 (see Fig. 5.11), but the precision on m j quicklydegrades at higher redshifts (Fenech Conti et al., 2017). Similar conclusions can be drawn from thephotometric redshift estimation, which becomes unreliable at high redshift when only using opticalbands (Hildebrandt et al., 2017). We see in our cross-correlation measurement a unique opportunity toplace a joint-constraint on these two quantities in the highest redshift bin, informed by our measurementsat lower redshift. We ignore the contribution from IA due to the high level of statistical noise in ourmeasurement. However, this will need to be included in similar analyses of upcoming surveys withhigher statistical precision.16Our approach is to fix the scaling term A to the value preferred by the zB ∈ [0.1,0.9] data, which welabel Alowz here for clarity, and to jointly fit for the mean shear bias and mean redshift distribution in thezB > 0.9 bin. Forcing A to this value in the high redshift bin provides constraints on 〈1+mhighz〉 and〈nhighz(z)〉, which we extract by varying these quantities in the predictions.The correction to the shear bias is trivial to implement as it scales linearly with A, so we simplywrite Ahighz = Alowz(1+ δm) = 0.95± 0.22 (from Table 5.2) and solve for δm. If this was the onlycorrection, we could write δm = Ahighz/Alowz− 1 = 0.38± 0.44, which is consistent with zero but notwell constrained.Corrections to the photometric distribution can be slightly more complicated, since the full redshiftdistribution that enters our calculation is not simple, as seen in Fig. 5.1. There are a number ways inwhich we could alter the n(z) and propagate the effect onto the signal, e.g., by modifying the overallshape, the mean or the tail of the distribution. We opted for arguably the simplest prescription, whichconsists of shifting the n(z) along the z direction by applying the mapping z→ z+δz (thereby shifting〈nhighz(z)〉 by the same amount). We propagate this new n(z) through equation 5.1 and solve for valuesof δz that satisfy constraints on A. In this process, we allow δz to vary by up to 0.5, which is ratherextreme.Following the simple reasoning described for the shear calibration, we can see that if m was trusted atthe percent level in this high redshift bin, constraints on the redshift distribution could be simply derivedby computing Ahighz/Alowz =Cδz/Cfid = (1+δz). We therefore obtain the exact same constraints as forδm, namely: δz = 0.38±0.44. We place constraints on the (δm,δz) plane by requiring (1+δm)(1+δz) =Ahighz/Alowz, and present the 1σ constraints in Fig. 5.10. The data are still consistent with δm = δz = 0,but these two biases are not currently well constrained.We also show in Fig. 5.10 the results from the CκCMBκgal` estimator (red dashed lines), but these havea lower SNR hence are not included in the analysis. At first sight, the difference observed betweenthe results from the two estimators might appear worrisome. Given that these constraints on (δm,δz)are obtained from the same data, and that the only difference is the analysis method, it is justified toquestion whether we could use this measurement for precise self-calibration if two methods on the16One might well object that the uncertainty in IA modelling and its evolution is already larger than the uncertainty inshear calibration, and hence that our strategy is flawed to start with; instead, we should be placing simultaneous constraintson the photo-z, m-calibration and IA. A full MCMC approach will certainly be required in the future to disentangle theseeffects, exploiting their different shape dependence to break the degeneracy between these parameters. As an illustration ofthis strategy, however, we present a simple case here and assume no IA contamination in this rest of this section.113−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−0.500.511.522.5δmδzCκ CMB κ galξκ CMB γ tFigure 5.10: 1σ contour regions on the shear calibration correction δm and the redshift distributioncorrection δz in the bin zB > 0.9, from the cross-correlation measurements. Black and redlines correspond to constraints from ξ κCMBγt and CκCMBκgal` respectively. The multiple linespresent the results in three different cosmologies (fiducial is solid, KiDS-450 is dot-dashed,Planck is dashed), which are shown here to have a small impact on the constraints. Otherindependent measurements and improved image simulations could tighten the region ofallowed values. The upper red solid and dot-dashed lines perfectly overlap.same data give such different values for δm and δz. We recall that differences are expected, since bothtechniques are probing different scales, however, the calibration technique presented here is sensitive tothese differences. The calibration is only weakly sensitive to the fiducial cosmology adopted, as shownin Fig. 5.10 using different line styles.A significant improvement will come from future data sets (advanced-ACT, SPT-3G, LSST, Euclid),in which the noise will be much lower, allowing for more accurate measurements of ξ κCMBγt and CκCMBκgal`to start with. In addition, including other measurements in this self-calibration approach will greatlyenhance the achievable precision. For example, one could measure the galaxy-galaxy lensing signalfrom the same KiDS-450 source galaxies, using, e.g., the GAMA galaxies as lenses (van Uitert et al.,2016), selecting the sources in the same tomographic bin (i.e., zB ∈ [0.1,0.9] and zB > 0.9). Fixing thecosmology from the low redshift bin, one could then similarly constrain (δm,δz) in the high redshiftbin. The idea here is that the trend can be made opposite to that seen in Fig. 5.10: an increase in δzpushes the sources away from the lenses, which, depending on the geometry, could reduce the signal.To compensate for this, the m-calibration would need to increase as well. In such a setup, the preferredregion in parameter space would inevitably intersect with ours, and exploiting this complementaritymight lead to competitive constraints. Further investigation of this combined measurement will beexplored in upcoming work.We are aware that our bi-linear modelling of the m and n(z) calibration is an over-simplificationof our knowledge (and uncertainty) about these quantities in the highest redshift bin, and one couldenvision improving this strategy in the future. For instance, the high-redshift objects are often thehardest to measure spectroscopically, hence there are greater chances that the DIR method fails at higher114−0.4 −0.2 0 0.2 0.4−0.6−0.4−δmδzA KiDSIAA KiDSA fidImage SimBootstrapFigure 5.11: 1σ contour regions on the shear calibration correction δm and the redshift distributioncorrection δz derived from the zB ∈ [0.1,0.9] measurement of A in three different cases.Results from Afid (no-IA) are shown in solid black, results from AKiDS (no IA) are shownin solid blue, and results from AKiDS with 10% IA are shown in solid red. The pair of solidhorizontal lines show the region of δm values allowed from image simulations, while the pairof dashed vertical lines show the region of δz values allowed from bootstrap resampling then(z).redshifts. To capture this effect, instead of shifting the n(z), one could modify only the high-redshift tail,moving 1%, 5% or 10% of our source galaxies from (very) low redshifts to z >1, propagating the effecton the signal, and use our measurement of A to constrain the fraction of such ‘missing’ high-redshiftgalaxies. However, given the size of our error bars, it is not clear that we would learn more from thisapproach at the moment.This situation will improve significantly with future CMB and galaxy surveys. According to Schaanet al. (2017), the lensing data provided by a Stage-4 CMB experiment, combined with 10 tomographicbins for LSST, will enable an m-calibration that is accurate to better than 0.5%. This comes frommarginalising over a number of nuisance parameters that unfortunately does not include catastrophicphotometric redshift outliers, so the actual accuracy will likely degrade compared to this impressivebenchmark. Nevertheless, this is an avenue that is certainly worth exploiting with the upcoming data.The zB < 0.9 redshift data in the KiDS survey has been calibrated on image simulations whoseprecision on δm largely surpasses that of the cross-correlation technique presented in this section. Fig-ure 5.11 shows the 1σ constraints in the (δm,δz) plane in the zB ∈ [0.1,0.9] bin assuming the fiducialcosmology without IA (black), the KiDS-450 cosmology without IA (blue), and the KiDS-450 with10% IA (red), consistent with both the HT-IA and fred-IA models. For comparison, the 1% precision onδm obtained from image simulations and the 3% precision on δz obtained from bootstrap resampling then(z), described in Section 5.5.4, are shown as the pairs of horizontal and vertical lines, respectively. Forthese redshifts at least, the measurement provides interesting constraints on the cosmology, IA and δz,but not on the m-calibration.1155.6 ConclusionsWe perform the first tomographic lensing-lensing cross-correlation by combining the Planck 2015 lens-ing map with the KiDS-450 shear data. Our measurement is based on two independent estimators, thePOLSPICE measurement of CκCMBκgal` , and the configuration-space measurement of ξκCMBγt(ϑ). The twotechniques agree within 1σ in all tomographic bins, although the former exhibits a lower signal to noiseratio.We compare our tomographic results against a two-dimensional lensing analysis of a single broadredshift bin (zB ∈ [0.1,0.9]), and fit the measured amplitude of the signal with a single multiplicativeparameter A that scales the predictions. We obtain Afid = 0.69± 0.15 in our fiducial cosmology, andshow that the constraints on the (σ8,Ωm) plane are consistent with the flat ΛCDM Planck cosmologyat the 95% level, with APlanck = 0.68±0.15, and with all previous results (Hand et al., 2015; Liu et al.,2015; Singh et al., 2017, K16 & HD16). The KiDS-450 cosmology is preferred, however, and in thiscase we obtain AKiDS = 0.86±0.19.Photometric redshifts have been examined carefully and are unlikely to be affecting these resultssignificantly (< 8% effect on the signal), unless the spectroscopic sample that is used to estimate the n(z)distribution suffers from significant sampling variance. Multiplicative shear calibration is also highlyunlikely to be affecting A, since it is known to be accurate at the percent level over the redshift rangethat enters our cosmological measurement. However, including different models of intrinsic alignment,massive neutrinos and baryon feedback in the predictions all affect the signal by tens of percent, pushingthe recovered A to higher values.Fixing the cosmology to that favoured by our low-redshift measurements (zB < 0.9), we calibratethe high-redshift (zB > 0.9) photometric n(z) and the multiplicative shear calibration, which are notrobustly constrained. We find that the high redshift data are consistent with no residual systematics,but that these are still allowed and only weakly constrained. Improved results on this high-redshiftcalibration will come in the future from larger data sets, from improved image simulations and from thecombination with other independent measurements.Tomographic measurements such as that presented in this chapter are insensitive to galaxy bias,and hence open the possibility of obtaining cosmological constraints from measurements of the growthfactor. Upcoming and future lensing surveys will provide excellent opportunities for combining probesand improving their cosmological constraints.116Chapter 6Concluding remarksThe fundamental question underlying all work presented in this thesis is the nature of dark matter. Thework explores different facets of this question, relying on the cross-correlation of gravitational lensingwith other cosmological probes.In Chapter 2 we investigate the microscopic nature of dark matter by measuring the cross-correlationbetween gamma rays from Fermi-LAT and weak gravitational lensing from CFHTLenS, RCSLenS,and KiDS. We do not observe a cross-correlation signal and use this non-detection to constrain themass, annihilation cross-section, and decay rate of WIMP dark matter. Accounting for astrophysicalsources of gamma rays and assuming strong clustering of dark matter at small scales, we are able toexclude the thermal relic cross-section for dark matter particle masses of mDM. 20 GeV. This constraintis comparable to other analyses of the extra-galactic gamma-ray flux, such as those of its isotropicintensity energy spectrum (Ackermann et al., 2015d) or its auto-power spectrum (Fornasa et al., 2016).Exclusion limits derived from local probes, such as dSphs, are stronger, however (Ackermann et al.,2015c). However, since our analysis is based on a cross-correlation with lensing, it probes differentstructures and is less affected by systematics. The work also presented the first use of tomography in aweak lensing cross-correlation, pushing the boundaries of this versatile technique.Baryon physics can significantly affect the matter distribution, especially at small scales (e.g., vanDaalen et al., 2011). Since gravitational lensing probes the total matter distribution, gravitational lensingmeasurements are affected by these baryonic processes (Semboloni et al., 2011; Harnois-De´raps et al.,2015b). It is therefore important to understand the behaviour of baryonic matter if we wish to make solidinference on the dark matter distribution from gravitational lensing. Chapter 3 presents the analysis ofthe cross-correlation of the thermal Sunyaev-Zeldovich (tSZ) effect, a direct tracer of hot, diffuse gas,and weak gravitational lensing from RCSLenS. A cross-correlation signal is detected at 8σ significance.The measured signal is then compared to predictions from the cosmo-OWLS suite of hydrodynamicalsimulations, where we find that the data prefer models with significant active galactic nuclei (AGN)feedback. The scatter of the simulations and the limited range of angular scales we have access to in thedata currently hide the scale-dependence of the different feedback models, making the effect of AGNfeedback degenerate with that of σ8. However, upcoming data and simulation products will be able tobreak this degeneracy and produce much tighter constraints using this technique.117The modelling of the cross-correlations signals uses a number of approximations to connect thethree-dimensional matter distribution to the two-dimensional angular power spectrum, such as the Bornapproximation and neglecting other higher order terms of the gravitational potential. In Chapter 4 weexamine these approximations in detail for the case of the cross-correlation of lensing and tSZ. Weconsider terms up to fourth order in the potential, including a new term due to rotation in the reducedshear. The standard second order expressions for the angular power spectrum are found to be sufficientlyaccurate for current and upcoming surveys. Certain third-order effects might become important at verysmall scales in future surveys, such as cross-correlations with a tSZ map from a CMB Stage 4 (Abazajianet al., 2016) experiment.Chapter 5 studies the tomographic cross-correlation of CMB lensing from Planck with galaxy lens-ing from KiDS. The tomographic measurement allows the analysis of the growth of structure, whichshows no deviation from the prediction based on a Universe made of cold dark matter and dark energy.Furthermore, we search for residual systematics in the galaxy lensing data. We find no evidence formultiplicative bias of the ellipticity estimate nor of errors in the estimate of the source distribution. Withcurrent data, the constraints on these biases from cross-correlation are still significantly weaker thanthose from direct methods (e.g, Fenech Conti et al., 2017; Hildebrandt et al., 2017), but future data setswill be able provide comparable and complementary constraints.6.1 Future prospectsAll analyses presented in this thesis are based on cross-correlation between essentially full-sky mapsand relatively small regions of weak lensing data. The small sky fraction of the weak lensing datasets – combined they reach about 1000 deg2 or 2.5% of the sky – restrict the power of these analyses.However, it also demonstrates the potential of these techniques for next-generation lensing surveys,such as full area KiDS, DES, LSST, and Euclid. For example, extrapolating our measurement of thecross-correlation between gamma rays and lensing suggests that a 4000 deg2 lensing survey with KiDScharacteristics would be able to detect a cross-correlation with astrophysical gamma-ray sources at3σ significance. A cross-correlation between full-depth DES and 10 years of Fermi-LAT data couldpotentially detect a dark matter gamma-ray signal at & 5σ significance (Camera et al., 2015).In the analysis of the cross-correlation between gravitational lensing and gamma rays the astro-physical contribution to the extra-galactic gamma-ray flux was considered a contaminant to our primarysignal, the gamma-ray flux from dark matter annihilations and decays. However, the astrophysical sig-nal is interesting in its own right. It is believed to be dominated by blazars – AGN with their jets pointedtowards us – and thus probes similar physical processes to those considered in Chapter 3. High-energyemissions such as gamma rays could therefore be used as an additional lever to calibrate simulations.The different cross-correlations have been analysed in isolation so far. Jointly analysing multiple,complementary probes is expected to break degeneracies and ensures that the different measurementsare consistent. In fact, a joint analysis of the three cross-correlations between galaxy lensing, tSZ, andCMB lensing is currently being performed by the author of this thesis. The primary aims are to constrainbaryon physics and the mass of the neutrinos. Since baryon physics affects the matter distribution in118the non-linear regime, the amplitude and scale-dependence of baryon physic effects on galaxy lensingand CMB lensing is not the same. The tSZ effect, by probing the gas directly, exhibits a differentdependence on baryonic processes again. The effects of massive neutrinos and cosmological parameters,such as σ8, on the matter distribution, and hence lensing, differs from that on the tSZ effect as well. Byexploiting the different effects of baryon physics, massive neutrinos, and cosmological parameters onthe different probes and their cross-correlations, we will be able to break the degeneracies we observedin the analyses of the single cross-correlations by themselves. In a similar vein, the joint analysiswill allow us to constrain systematics, such as m-corrections, redshift distribution uncertainties, andintrinsic alignment more efficiently, thus making the analysis and its inferences more robust. Includingupcoming CMB data sets, such as from ACTPol (Thornton et al., 2016) will reduce the noise in thereconstruction of the lensing of the CMB and improve the resolution of the tSZ map at small scales,increasing the overall SNR of the measurements and allowing us to further break degeneracies throughaccess to smaller scales. The joint analysis of the cross-correlations between galaxy lensing, tSZ, andCMB lensing can be augmented by the auto-correlations of the three probes and inclusions of externaldata sets at the likelihood level, such as the primary CMB and BAO data, allowing us to further increaseour understanding of baryon physics, massive neutrinos, and cosmology, while tightening constraintson systematics.If one only cares about cosmological parameters, such as the dark energy equation of state, a com-plimentary approach to modelling baryon physics is to marginalise over a suitable set of parametersdescribing the effects of baryonic processes (Zentner et al., 2008, 2013; Eifler et al., 2015). However,this approach also benefits from including probes that are sensitive to baryon physics, as they allow usto find better sets of parameters describing baryonic effects and hence tighten their priors.A recurring difficulty in the works presented in this thesis has been the accurate estimation of thedata covariance. While the covariance of the power spectrum in the linear regime is diagonal and onlydepends on power spectra, non-linear processes cause the different modes to couple and the trispectrumcontribution becomes non-negligible. Modelling the trispectrum is difficult, considering that findingaccurate models for the power spectrum is often a formidable task in itself. An alternative approachis to create many realisations of the data from simulations and estimate the data covariance directlyfrom the covariance of these realisations. To reduce the noise in the covariance, large numbers ofrealisations are required (Hartlap et al., 2007; Taylor et al., 2013, 2014). The computational cost canbe prohibitively expensive when hydrodynamical simulations are required, like in the case of the tSZeffect. Reducing the size of the data vector through data compression schemes, such as Karhunen-Loe´vemethods (Tegmark et al., 1997), can reduce the number of required realisations but the effectiveness ofthis approach is still a topic of ongoing research. A third approach is to rely on internal covarianceestimators, such as the jackknife and bootstrap methods. These methods are computationally cheapand require no modelling. However, the simple implementations used in cosmology are known to yieldbiased covariance estimates. Developing methods to correct for these biases would make it possibleto take full advantage of the benefits of internal covariance estimators and is therefore a worthwhileendeavour.119BibliographyK. N. Abazajian, P. Adshead, Z. Ahmed, S. W. Allen, D. Alonso, K. S. Arnold, C. Baccigalupi, J. G.Bartlett, N. Battaglia, B. A. Benson, C. A. 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D, 87(4):043509, Feb. 2013. doi:10.1103/PhysRevD.87.043509. → pages 119141Appendix ASupplementary material to Chapter 2A.1 Fourier-space estimator performanceTo check the ability of the power spectrum estimator based on the integration of the tangential shearcorrelation function, given by Eq. (2.12), to recover the true underlying power spectrum, we test it ona suite of mock simulations. We wish to make a generic test of the accuracy of the power spectrumestimation. To this end, we compare the auto-spectrum of the convergence with the estimated cross-spectrum between the convergence and the shear, which are expected to yield the same result on the flatpatches of the simulations. This is analogous to the cross-spectrum of the gamma rays and shear in theanalysis of Chapter 2 but easier to handle, as high-resolution simulation products for convergence andshear are readily available.The simulation products we use are part of the Scinet LIght Cone Simulation suite(Harnois-De´raps et al., 2015a, SLICS hereafter), which consist of 930 realizations of lens-ing data over 10◦ × 10◦ patches in a WMAP9+SN+BAO cosmology ({ΩM,ΩΛ,Ωb,σ8,h,ns} ={0.2905,0.7095,0.0473,0.826,0.6898,0.969}). The convergence and two shear components are con-structed by ray-tracing up to 18 density planes between redshift zero and 3, and finally mapped on to77452 pixels (see the SLICS reference for details about how this is implemented numerically). For ourparticular setup, we use the maps constructed while assuming that the galaxy sources are all placed atredshift 0.582. This is of course not representative of the real galaxy distribution of the data, but closelymatches the mean of the distribution, which is sufficient for the purpose of calibration.For the verification of our estimator in Eq. (2.12), we use a subset of 100 realizations. The conver-gence and shear maps are cropped to 77002 pixels and then down-sampled by a factor of 10 to closerresemble the pixel size encountered in the gamma-ray analysis.We measure the tangential shear correlation function between the convergence and shear maps usingthe same binning scheme as the gamma-ray cross-correlation measurement, i.e., 300 linearly spacedbins between 1 and 301 arcmin. The power spectrum estimated using Eq. (2.12) is then expected toagree with the auto-power spectrum of the convergence map Cκκ` . The power spectra measured on thesimulations are shown in Fig. A.1. For the scales of interest in the analysis of Chapter 2 the estimator1420.`(`+1)/2piC`×10−5ξ → C`ξ → C`, band powerTrue C`True C`, band power200 400 600 800 1000 1200 1400`−0.4−`−pC`)/pC`1 LOS1 LOS, band power100 LOS, band powerFigure A.1: Top: power spectrum estimated using Eq. (2.12) (ξ →C`, dash-dotted blue), ξ →C`band power (solid blue), true power spectrum from SLICS (dotted magenta), and true bandpower from SLICS (dashed magenta) for one line-of-sight. Bottom: difference betweenestimated and true power spectrum from SLICS (light solid blue) for one line-of-sight, dif-ference between the estimated and true band power from SLICS for one line-of-sight (hashedblue) and 100 lines-of-sight (solid green).recovers the power spectrum to within 5 percent on individual line-of-sights, which is within the error onthe mean per `-bin of the true power spectrum. The agreement is within 1 per cent for 100 lines-of-sight,which is less than the error on the mean of the 100 true power spectra, showing that the fluctuations seenon individual lines-of-sight average out.One caveat is that the range of integration in Eq. (2.12) is formally from 0 to infinite angular sep-aration. By restricting the integration to some finite range ϑmin to ϑmax, the resulting power spectrumestimate can become biased.To estimate the effect of restricting the angular range of the two-point correlation function on thepower spectrum estimate we produce a high-resolution measurement of the convergence power spectrumCκκ` . Using the relation in Eq. (2.2) between the power spectrum Cκκ` and the tangential shear correlationfunction ξ κγt (ϑ), we compute a theory estimate ξ κγtth (ϑ) from the measured high-resolution convergencepower spectrum. Alternatively, we could also have used an analytical model for the power spectrum orcorrelation function. However, since we have access to the true power spectrum from SLICS and in1430.`(`+1)/2piC`×10−5ξ → C`, band powerTrue C`, band powerTrue C`, band power, corrected500 1500 2500 3500 4500 5500`−0.5−0.4−0.3−0.2−`−pC`)/pC`1 LOS band power 1 LOS band power, correctedFigure A.2: Top: band power spectrum estimated using Eq. (2.12) (ξ →C`, solid blue), true bandpower from SLICS (dashed magenta), and true band power from SLICS including the effectof finite ϑmin in Eq. (A.1)(dotted magenta). Bottom: difference between estimated bandpower and true band power from SLICS (solid green) and adjusted for finite ϑmin (hashedblue).order to make the comparison between the different power spectrum estimates as direct as possible, wechose to minimize the amount of external information.Finally, to estimate the effect of restricting the range of integration we compute the corrections termsCϑmin` = 2pi∫ ϑmin0dϑ ′ ϑJ2(`ϑ ′)ξκγtth (ϑ′) (A.1)andCϑmax` = 2pi∫ ∞ϑmaxdϑ ′ ϑJ2(`ϑ ′)ξκγtth (ϑ′) . (A.2)The effect of the minimum angular separation ϑmin is a suppression of power at small scales, restrictingthe range of scales where the power spectrum estimate is unbiased. However, by forward modelling,i.e., accounting for the effect of the minimum angular separation when comparing the measurementsto models, the effective range can be increased significantly. This can be seen in Fig. A.2, where144accounting for Cϑmin` increases the range of validity from `≈ 2000 to `≈ 6000. On the scales consideredin the analysis of Chapter 2 (`≤ 1500) the effect of a finite ϑ is negligible and no forward modelling ofthis effect was conducted.A finite maximum angular separation ϑmax does not lead to a systematic bias in the power spectrumestimation like the effect of ϑmin. The oscillatory nature of the estimated power spectrum requires theuse of band powers, however. The width of the bins results in an effective lower limit on the scalesthat can be estimated. For fixed ϑ ′, the Bessel function in Eq. (2.12) oscillates with period of 2piϑ ′ . Theshortest period that can be probed is therefore around 2piϑ ′max . Requiring two to three periods per `-bin,the minimum reliable bin width for a maximum angular separation of ϑmax = 301 arcmin is therefore∆` ≈ 200. The bin-width chosen in Chapter 2 is ∆` = 260 and can thus be assumed to yield a reliableestimate of the power spectrum.The accuracy of the estimator is thus sufficient for the analysis in Chapter 2. Applications to higherSNR probes, such as cosmic shear, or future surveys, such as LSST, would require further testing, how-ever. At large scales, the effect of the finite maximum angular separation ϑmax could become importantenough to require inclusion in the forward modelling. The effect of using the flat-sky approximationalso needs to quantified and might warrant a reformulation to a full-sky formalism.145Appendix BSupplementary material to Chapter 3B.1 Extra considerations in κ-map reconstructionIn the following, we describe the set of extra processing steps we have performed in our κ-map re-construction pipeline to improve the SNR of our cross correlation measurements. These include theselection function applied to the lensing shear catalogue, adjustments in the reconstruction process, andproper masking of the contamination in the tSZ y maps.Magnitude SelectionOne of the parameters that we can optimize to increase the SNR is the magnitude selection function withwhich we select galaxies from the shear catalogue (and then make convergence maps from). This is nottrivial, as it is not obvious whether including faint sources with a noisy shear signal would improve oursignal or not. To investigate this, we apply different magnitude cuts to our shear catalogue and computethe SNR of the correlation function measurement in each case.In Fig. B.1, we compare the correlation functions from three different r-band magnitude cuts:21–23.5; 18–24; and ≥ 18 (all of the objects). We find that the variations in the mean signal due todifferent magnitude cuts are relatively small. However, there is still considerable difference in the scat-ter around the mean signal, which results in different SNR for the cuts. We consistently found thatincluding all the objects (no cut) leads to a higher SNR.Impact of smoothingAnother factor that can change the SNR of the measurements is the smoothing kernel we apply to thelensing maps. Note that in our analysis, the resolution of the cross-correlation (the smallest angularseparation) is limited by the resolution of the tSZ maps, which matches the observational beam scalefrom the Planck satellite (FWHM= 10 arcmin). On the other hand, lensing maps could have a muchhigher resolution and the interesting question is how the smoothing scale of the lensing maps affects theSNR.As described before, the configuration-space y–γt cross-correlation works at the catalogue level14620 40 60 80 100 120 140 160 180ϑ [arcmin]−ξy−κ(ϑ)×10−9Mag 21.5-23Mag 18-24Mag >1820 40 60 80 100 120 140 160 180ϑ [arcmin]01234567ξy−γt(ϑ)×10−10Mag 21.5-23Mag 18-24Mag >18Figure B.1: Impact of different magnitude cuts on the y–κ (left) and y–γt (right) cross-correlationsignals. Including all the sources in the lensing surveys yields the highest SNR.without any smoothing involved. However, in making the convergence maps, we apply a smoothingkernel as described in van Waerbeke et al. (2013). One has therefore the freedom to smooth the con-vergence maps with an arbitrary kernel. We consider three different smoothing scales and evaluate theSNR of the cross-correlations.Figure B.2 demonstrates the impact of applying different smoothing scales (FWHM = 3.3, 10 and16.5 arcmin) to the convergence maps used for the y–κ cross-correlation. Note that while narrowersmoothing kernels results in a higher cross-correlation amplitude, the uncertainties also increase andit lowers the SNR. We concluded that smoothing the maps with roughly the same scale as the y maps(FWHM ≈ 9.5 arcmin) leads to the highest SNR.Masks on the y mapsSince we work with tSZ maps provided by the Planck collaboration, there is a minimal processingof the y maps for our analysis. We apply the masks provided by the Planck collaboration to removepoint sources and Galactic contamination. Note that the Galactic mask does not significantly affectour measurements since all the RCSLenS fields are at high enough latitude. Cross-correlations are notsensitive to uncorrelated sources such as Galactic diffuse emission and point sources either. We havechecked that our signal is robust against the masking of the point sources (see Fig. B.3).B.2 Null tests and other effectsWe have performed several consistency checks to verify our map reconstruction procedures and therobustness of the measurements. As mentioned before, an advantage of a cross-correlation analysisis that those sources of systematics that are unrelated to the measured signal will be suppressed inthe measurement. This is particularly useful in the case of RCSLenS. As described below, there areresidual systematics in the RCSLenS shear data (see Hildebrandt et al., 2016, for details). It is therefore14720 40 60 80 100 120 140 160 180ϑ [arcmin]−ξy−κ(ϑ)×10−9FWHM 3.3FWHM 10FWHM 16.7Figure B.2: Impact of varying the smoothing of the convergence maps on the y–κ cross-correlationsignal. We apply three different smoothing scales of FWHM = 3.3, 10 and 16.7 arcmin dur-ing the mass map making process. While smaller scales result in a higher cross-correlationsignal, the SNR is best for a smoothing scale of the same order as that of the y maps (FWHM= 10 arcmin).20 40 60 80 100 120 140 160 180ϑ [arcmin]−ξy−κ(ϑ)×10−9refno point-source mask20 40 60 80 100 120 140 160 180ϑ [arcmin]01234567ξy−γt(ϑ)×10−10refno point-source maskFigure B.3: The impact of masking point sources in the y map on the y–κ (left) and y–γt (right)cross correlation analysis. The measurements are fairly robust against such contamination.important to check if these systematics contaminate our cross-correlation. We start with a descriptionof lensing B-mode residuals in RCSLenS data.Lensing B-mode residualsIn the absence of residual systematics, the scalar nature of the gravitational potential should lead to avanishing convergence B-mode signal. As one of the important systematic checks in a weak lensingsurvey, one should investigate the level of the B-mode present in the constructed mass maps.To check for B-mode residuals in the RCSLenS data, we first create a new shear catalogue by rotating1480 20 40 60 80 100 120 140 160ϑ[arcmin]−ξ(ϑ)×10−6κB − κBκE − κBFigure B.4: The stacked B-mode residual from the RCSLenS fields represented through the auto-correlation function, after subtracting the statistical noise contribution. The signal is notconsistent with zero due to residual systematics in the shape measurements. The κE − κBcross-correlation is also shown which is consistent with zero.each galaxy in the original RCSLenS catalogue by 45◦ in the observation plane. This is equivalent toapplying a transformation of shear components from (γ1,γ2) to (−γ2,γ1) (Schneider et al., 1998). Wethen follow our standard mass map making procedure to construct B-mode convergence maps, κobsB ,from the new catalogue. Similar to the original maps, these maps are noisy and consists of the trueunderlying convergence, κB, and additional statistical noise, κran:κobsB = κB+κran. (B.1)It is, therefore, necessary to distinguish between the two components when searching for residual B-modes.To estimate κran, we produce many ‘noise’ catalogue where this time the orientations of galaxies arerandomly changed (these are essentially the same maps that are used to construct the covariance ma-trix as described in Section 3.4). The constructed mass maps from these catalogue would only containstatistical noise. We estimate an average statistical noise auto-correlation function, ξ¯κran , for each RC-SLenS field by averaging over the auto-correlation function from the random mass map realizations ofthe field. Finally, we estimate the residual B-mode signal in each of the RCSLenS fields by subtractingthe statistical noise contribution computed for that field from the observed auto-correlation function.Figure B.4 shows the weighted average of the residual B-mode correlation function computed fromthe 14 RCSLenS fields after subtracting the contribution from statistical noise. The error bars representthe error on the mean value in each angular bin. Note that there is an excess of residual B-modes atangular separations of ≤ 40 arcmin. Independent analysis of projected 3D shear power spectrum alsoconfirms presence of excess residual B-mode signal at the corresponding scales (Hildebrandt et al.,2016), consistent with our finding. The existence of such residual systematics could be problematic forour studies and needs to be checked as we describe in the following.149In Fig. B.4, we show E- and B-mode mass map cross-correlation from RCSLenS. The cross-correlation signal is consistent with zero which shows that any possible leakage of the systematic B-mode residuals to E-mode does not correlate with the true E-mode signal.Residual tSZ-lensing systematic correlation0 20 40 60 80 100 120 140 160ϑ[arcmin]−0.4−ξy−κ(ϑ)×10−9κEκB50 100 150ϑ [arcmin]−10123456ξy−γt(ϑ)×10−10γtγxFigure B.5: Summary of the null tests performed on the y–κ (left) and y–γt (right) estimators. Thered squares show the B-mode κ (right) and γ× cross-correlations which are consistent withzero as expected (the blue circles show the E-mode κ (right) and γt cross-correlations forcomparison). The null tests are validated in all cases confirming that the level of contami-nating systematics in RCSLenS are under control in the cross-correlation analysis.In the following, we demonstrate that while there is significant B-mode signal in the RCSLenS data,the lensing-tSZ cross-correlation signal is not contaminated. This serves as a good example of howcross-correlating different probes can suppress significant systematic residuals and make it useful forfurther studies.As the first step, we cross-correlate the y maps with the random noise maps constructed for eachRCSLenS field. We computed the mean cross-correlation and the error on the mean from these randomnoise maps. Consistency with zero insures that there is not any unexpected correlation between the ysignal in the absence of a true lensing signal and insures that the field masks do not create any artefacts.As the next step, we correlate the y maps with the constructed κB maps. This cross-correlationshould also be consistent with zero to ensure that there is no unexpected correlation between the tSZsignal and the systematic lensing B-mode. Note that we can perform a similar consistency check usingthe shear data instead. The analogue to the κB mode for shear is the cross (or radial) shear quantity, γ×,defined asγ×(θ) =−γ1 cos(2φ)+ γ2 sin(2φ), (B.2)which can be constructed by 45◦ rotation of galaxy orientation in the shear catalogue. With this estima-tor, we expect the y–γ× cross correlations to be consistent with zero as another check of systematics inour measurements.150Fig. B.5 summarizes the null tests described above. In the left panel, cross-correlations of y withreconstructed κB maps are shown with the y–κE curve over-plotted for comparison. Correlation with κBmaps slightly deviates from zero at smaller scales due to residual systematics, but is still insignificantconsidering the high level of bin-to-bin correlation. In the right panel, we show cross-correlation with γ×with the y–γt curve over-plotted for comparison. We do not see any inconsistency in the y–γ× correlation.Both estimators are also perfectly consistent with zero when cross-correlated with random maps.We therefore conclude that the systematic residuals are sufficiently under control and do not affectour measurements at a measurable level for this data set.151Appendix CSupplementary material to Chapter 4C.1 Fourier space identitiesAdapting the notation of Dodelson et al. (2005a), we define the 2D Fourier transform on the planeperpendicular to the line-of-sight asΦ( fK(χ)~θ ,χ) =∫ d2`′(2pi)2φˆ(~`′,χ)ei~`′~θ , (C.1)with the angular transform of the field Φ given byφˆ(~`,χ) =∫ dk32pi1fK(χ)2Φˆ(~`fK(χ),k3)eik3χ . (C.2)The higher order expressions for y and κ involve products of the potential Φ. In Fourier space, theseproducts become convolutions. For two fields F and G we havê[FG](~`) = [Fˆ ∗ Gˆ](~`) = ∫ d2`′(2pi)2Fˆ(~`′)Gˆ(~`−~`′) . (C.3)This generalizes straightforwardly to the case of three fields F, G, and K as[̂FGK](~`) =[Fˆ ∗ Gˆ∗ Kˆ](~`) = ∫ d2`′d2`′′(2pi)4Fˆ(~`′)Gˆ(~`′′)Kˆ(~`−~`′− ~`′′) . (C.4)The two-point correlation function of the fields φˆ(~`,χ) is directly related to the power spectrum of thepotential Φ. Assuming homogeneity, isotropy, and using the Limber approximation (Limber, 1953;Kaiser, 1992), i.e., assuming that |~`|  k3, thus justifying neglecting the longitudinal modes, the two-152point function can be written as〈φˆ(~`1,χ1)φˆ(~`2,χ2)〉= (2pi)2δD(χ1−χ2)δD(~`1+~`2)Cφφ`= (2pi)2δD(χ1−χ2)δ 2D(~`1+~`2)fK(χ1)2PΦ(|~`1|fK(χ1),χ1).(C.5)Similarly, the three-point function of φˆ(~`,χ) is related to the bispectrumBΦ(~k1,~k2,~k3,χ) by〈φˆ(~`1,χ1)φˆ(~`2,χ2)φˆ(~`3,χ3)〉= (2pi)2δD(χ1−χ2)δD(χ1−χ3)δ 2D(~`1+~`2+~`3)fK(χ1)4×BΦ(|~`1|fK(χ1),|~`2|fK(χ1),|~`3|fK(χ1),χ1). (C.6)The four-point function can be expressed in terms of the trispectrum as〈φˆ(~`1,χ1)φˆ(~`2,χ2)φˆ(~`3,χ3)φˆ(~`4,χ4)〉= (2pi)2δD(χ1−χ2)δD(χ1−χ3)δD(χ1−χ4)×δ 2D(~`1+~`2+~`3+~`4)fK(χ1)6×TΦ(|~`1|fK(χ1),|~`2|fK(χ1),|~`3|fK(χ1),|~`4|fK(χ1),χ1).(C.7)Using the definition of the Fourier transform (C.1), partial derivatives with respect to comoving trans-verse coordinates can be written asΦ,i1...iN ( fK(χ)~θ ,χ) =∫ d2`′(2pi)2iNfK(χ)N`′i1 . . . `′iN φˆ(~`′,χ)ei~`′~θ . (C.8)C.2 Convergence - shear relationIn this appendix we show that the relation (4.27) holds even beyond first order, justifying the use ofthe convergence as the fundamental quantity instead of the shear. The second-order expressions of theconvergence and shear due to Born approximation and lens-lens couplings in Fourier space areκˆ(2)std (~`,χ) =−2∫ χ0dχ ′∫ χ ′0dχ ′′K(χ,χ ′)K(χ ′,χ ′′)fK(χ ′)2 fK(χ ′′)2×∫ d2`′(2pi)2|~`′||~`|cos(φ`−φ`′)[~`′(~`−~`′)]φˆ(~`′,χ ′)φˆ(~`−~`′,χ ′′)(C.9)153and(γˆ(2)std )I(~`,χ) =−2∫ χ0dχ ′∫ χ ′0dχ ′′K(χ,χ ′)K(χ ′,χ ′′)fK(χ ′)2 fK(χ ′′)2×∫ d2`′(2pi)2|~`′||~`|UI(~`,~`′)[~`′(~`−~`′)]φˆ(~`′,χ ′)φˆ(~`−~`′,χ ′′) ,(C.10)where the couplings UI are given byU1(~`,~`′) = cos(φ`+φ`′) , U2(~`,~`′) = sin(φ`+φ`′) . (C.11)Using the identityT I(~`)UI(~`,~`′) = cos(φ`−φ`′) , (C.12)where T I is given in (4.27), we have thus shown that κˆ(2)std = TI(γˆ(2)std )I .Generally, the relation does not hold anymore at third order. As we are only concerned with corre-lation functions in this work, it is sufficient to show that the relation holds within correlation functions,i.e., 〈yˆ(1)κˆ(3)std 〉 = 〈yˆ(1)T I(γˆ(3)std )I〉. To do so we first note that the third term in the third-order expressionfor the convergence (4.21) does not contribute to the correlation function under the Limber approxima-tion. This also applies to the third-order shear, since the line-of-sight integrals are the same for both theconvergence and the shear. The mode coupling terms of the angular cross-power spectra are〈yˆ(1)(~`1)κˆ(3)std (~`2)〉 ∝∫ d2`′(2pi)2(|~`2|2+2~`2~`′)|~`1|2[~`2~`′]2(C.13)and〈yˆ(1)(~`1)T I(~`2)(γˆ(3)std )I(~`2)〉 ∝ T I(~`2)∫ d2`′(2pi)2(|~`2|2TI(~`2)−2|~`2||~`′|UI(~`2,~`′))|~`1|2[~`2~`′]2. (C.14)Applying the two identities T I(~`)TI(~`′) = cos(2φ`′−2φ`) and (C.12) we see that the two above expres-sions are equal. We have thus proven that it is justified to use the convergence in cross-power spectrainstead of terms of the from T I(~`)γˆI(~`), up to third order. To second order, the relation κˆ = T I γˆI evenholds exactly.154C.3 Induced rotationLet S(~θ) be the surface brightness distribution of an extended source. The first and second moments ofthe brightness distributions are then defined as (Schneider et al., 1995)θ 0i =∫d2~θ θiS(~θ)∫d2~θ S(~θ), (C.15a)Qi j =∫d2~θ (θi−θ 0i )(θ j−θ 0j )S(~θ)∫d2~θ S(~θ). (C.15b)Following Seitz et al. (1997), we introduce the complex ellipticity parameterε =Q11−Q22+2iQ12Q11+Q22+2√Q11Q22−Q212. (C.16)For an elliptical source with semi-major and semi-minor axis a and b, rotated by an angle α with respectto a fixed coordinate system, the ellipticity parameter (C.16) is given byε =a−ba+be2iα . (C.17)The Jacobi map (4.3) relates an infinitesimal distance on the source plane to an infinitesimal distance onthe image plane by d~θS = A (~θS) d~θO. Assuming the source is sufficiently small such that the Jacobimap does not vary over the extend of the source, the second brightness moment (C.15b) of the sourceQS can be approximately related to the observed second brightness moment QO byQS =A QOA T . (C.18)We generalize previous work by allowing A (~θ) to have an anti-symmetric part. This anti-symmetriccontribution ω can be thought of as a rotation induced by lens-lens coupling. Given an elliptical source,the observed ellipticity can be written asεO =g+ εS′1+g∗εS′, (C.19)where the generalized reduced shear g and rotated source ellipticity εS′ are given byg =γ1+ iγ21−κ+ iω , εS′ = εS e−2iϑ , tanϑ =ω1−κ . (C.20)The reduced shear now includes a contribution from the anti-symmetric term ω of the general Jacobimap. Furthermore, the source ellipticity is rotated by an angle ϑ . However, assuming the sources aredistributed isotropically, this rotation is not observable. In particular, the ensemble average 〈εO〉 = g,i.e., the observed ellipticity remains an unbiased estimator of the reduced shear despite the rotation ω .In the limit of a symmetric Jacobi map ω → 0 one recovers eq. (3.2) of Seitz et al. (1997).155Finally, we express the generalized reduced shear in vector notation to facilitate the use in section4.3.2. The two components areg1 =ℜ(g) =γ1(1−κ)+ γ2ω(1−κ)2+ω2 , g2 = ℑ(g) =γ2(1−κ)− γ1ω(1−κ)2+ω2 . (C.21)Because ω is necessarily of at least second order, the generalized reduced shear to third order can bewritten asgI =γI1−κ +R(ω)IJγJ +O(Φ4) , (C.22)where the matrix R(ω) is defined asR(ω)IJ =(0 ω−ω 0). (C.23)This can be understood as an infinitesimal rotation of the shear by an angle ω .156


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